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Theorem axtgpasch 27718
Description: Axiom of (Inner) Pasch, Axiom A7 of [Schwabhauser] p. 12. Given triangle π‘‹π‘Œπ‘, point π‘ˆ in segment 𝑋𝑍, and point 𝑉 in segment π‘Œπ‘, there exists a point π‘Ž on both the segment π‘ˆπ‘Œ and the segment 𝑉𝑋. This axiom is essentially a subset of the general Pasch axiom. The general Pasch axiom asserts that on a plane "a line intersecting a triangle in one of its sides, and not intersecting any of the vertices, must intersect one of the other two sides" (per the discussion about Axiom 7 of [Tarski1999] p. 179). The (general) Pasch axiom was used implicitly by Euclid, but never stated; Moritz Pasch discovered its omission in 1882. As noted in the Metamath book, this means that the omission of Pasch's axiom from Euclid went unnoticed for 2000 years. Only the inner Pasch algorithm is included as an axiom; the "outer" form of the Pasch axiom can be proved using the inner form (see theorem 9.6 of [Schwabhauser] p. 69 and the brief discussion in axiom 7.1 of [Tarski1999] p. 180). (Contributed by Thierry Arnoux, 15-Mar-2019.)
Hypotheses
Ref Expression
axtrkg.p 𝑃 = (Baseβ€˜πΊ)
axtrkg.d βˆ’ = (distβ€˜πΊ)
axtrkg.i 𝐼 = (Itvβ€˜πΊ)
axtrkg.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
axtgpasch.1 (πœ‘ β†’ 𝑋 ∈ 𝑃)
axtgpasch.2 (πœ‘ β†’ π‘Œ ∈ 𝑃)
axtgpasch.3 (πœ‘ β†’ 𝑍 ∈ 𝑃)
axtgpasch.4 (πœ‘ β†’ π‘ˆ ∈ 𝑃)
axtgpasch.5 (πœ‘ β†’ 𝑉 ∈ 𝑃)
axtgpasch.6 (πœ‘ β†’ π‘ˆ ∈ (𝑋𝐼𝑍))
axtgpasch.7 (πœ‘ β†’ 𝑉 ∈ (π‘ŒπΌπ‘))
Assertion
Ref Expression
axtgpasch (πœ‘ β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑉𝐼𝑋)))
Distinct variable groups:   𝐼,π‘Ž   𝑃,π‘Ž   π‘ˆ,π‘Ž   𝑋,π‘Ž   π‘Œ,π‘Ž   𝑍,π‘Ž   𝑉,π‘Ž   βˆ’ ,π‘Ž
Allowed substitution hints:   πœ‘(π‘Ž)   𝐺(π‘Ž)

Proof of Theorem axtgpasch
Dummy variables 𝑓 𝑖 𝑝 π‘₯ 𝑦 𝑧 𝑏 𝑣 𝑠 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axtgpasch.6 . 2 (πœ‘ β†’ π‘ˆ ∈ (𝑋𝐼𝑍))
2 axtgpasch.7 . 2 (πœ‘ β†’ 𝑉 ∈ (π‘ŒπΌπ‘))
3 df-trkg 27704 . . . . . . 7 TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(Itvβ€˜π‘“) / 𝑖](LineGβ€˜π‘“) = (π‘₯ ∈ 𝑝, 𝑦 ∈ (𝑝 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (π‘₯𝑖𝑦) ∨ π‘₯ ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (π‘₯𝑖𝑧))})}))
4 inss1 4229 . . . . . . . 8 ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(Itvβ€˜π‘“) / 𝑖](LineGβ€˜π‘“) = (π‘₯ ∈ 𝑝, 𝑦 ∈ (𝑝 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (π‘₯𝑖𝑦) ∨ π‘₯ ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (π‘₯𝑖𝑧))})})) βŠ† (TarskiGC ∩ TarskiGB)
5 inss2 4230 . . . . . . . 8 (TarskiGC ∩ TarskiGB) βŠ† TarskiGB
64, 5sstri 3992 . . . . . . 7 ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(Itvβ€˜π‘“) / 𝑖](LineGβ€˜π‘“) = (π‘₯ ∈ 𝑝, 𝑦 ∈ (𝑝 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (π‘₯𝑖𝑦) ∨ π‘₯ ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (π‘₯𝑖𝑧))})})) βŠ† TarskiGB
73, 6eqsstri 4017 . . . . . 6 TarskiG βŠ† TarskiGB
8 axtrkg.g . . . . . 6 (πœ‘ β†’ 𝐺 ∈ TarskiG)
97, 8sselid 3981 . . . . 5 (πœ‘ β†’ 𝐺 ∈ TarskiGB)
10 axtrkg.p . . . . . . . 8 𝑃 = (Baseβ€˜πΊ)
11 axtrkg.d . . . . . . . 8 βˆ’ = (distβ€˜πΊ)
12 axtrkg.i . . . . . . . 8 𝐼 = (Itvβ€˜πΊ)
1310, 11, 12istrkgb 27706 . . . . . . 7 (𝐺 ∈ TarskiGB ↔ (𝐺 ∈ V ∧ (βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 (𝑦 ∈ (π‘₯𝐼π‘₯) β†’ π‘₯ = 𝑦) ∧ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (π‘₯𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼π‘₯))) ∧ βˆ€π‘  ∈ 𝒫 π‘ƒβˆ€π‘‘ ∈ 𝒫 𝑃(βˆƒπ‘Ž ∈ 𝑃 βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ ∈ (π‘ŽπΌπ‘¦) β†’ βˆƒπ‘ ∈ 𝑃 βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 ∈ (π‘₯𝐼𝑦)))))
1413simprbi 498 . . . . . 6 (𝐺 ∈ TarskiGB β†’ (βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 (𝑦 ∈ (π‘₯𝐼π‘₯) β†’ π‘₯ = 𝑦) ∧ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (π‘₯𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼π‘₯))) ∧ βˆ€π‘  ∈ 𝒫 π‘ƒβˆ€π‘‘ ∈ 𝒫 𝑃(βˆƒπ‘Ž ∈ 𝑃 βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 π‘₯ ∈ (π‘ŽπΌπ‘¦) β†’ βˆƒπ‘ ∈ 𝑃 βˆ€π‘₯ ∈ 𝑠 βˆ€π‘¦ ∈ 𝑑 𝑏 ∈ (π‘₯𝐼𝑦))))
1514simp2d 1144 . . . . 5 (𝐺 ∈ TarskiGB β†’ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (π‘₯𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼π‘₯))))
169, 15syl 17 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (π‘₯𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼π‘₯))))
17 axtgpasch.1 . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝑃)
18 axtgpasch.2 . . . . 5 (πœ‘ β†’ π‘Œ ∈ 𝑃)
19 axtgpasch.3 . . . . 5 (πœ‘ β†’ 𝑍 ∈ 𝑃)
20 oveq1 7416 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ (π‘₯𝐼𝑧) = (𝑋𝐼𝑧))
2120eleq2d 2820 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ (𝑒 ∈ (π‘₯𝐼𝑧) ↔ 𝑒 ∈ (𝑋𝐼𝑧)))
2221anbi1d 631 . . . . . . . 8 (π‘₯ = 𝑋 β†’ ((𝑒 ∈ (π‘₯𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) ↔ (𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧))))
23 oveq2 7417 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ (𝑣𝐼π‘₯) = (𝑣𝐼𝑋))
2423eleq2d 2820 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ (π‘Ž ∈ (𝑣𝐼π‘₯) ↔ π‘Ž ∈ (𝑣𝐼𝑋)))
2524anbi2d 630 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ ((π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼π‘₯)) ↔ (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼𝑋))))
2625rexbidv 3179 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼π‘₯)) ↔ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼𝑋))))
2722, 26imbi12d 345 . . . . . . 7 (π‘₯ = 𝑋 β†’ (((𝑒 ∈ (π‘₯𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼π‘₯))) ↔ ((𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼𝑋)))))
28272ralbidv 3219 . . . . . 6 (π‘₯ = 𝑋 β†’ (βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (π‘₯𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼π‘₯))) ↔ βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼𝑋)))))
29 oveq1 7416 . . . . . . . . . 10 (𝑦 = π‘Œ β†’ (𝑦𝐼𝑧) = (π‘ŒπΌπ‘§))
3029eleq2d 2820 . . . . . . . . 9 (𝑦 = π‘Œ β†’ (𝑣 ∈ (𝑦𝐼𝑧) ↔ 𝑣 ∈ (π‘ŒπΌπ‘§)))
3130anbi2d 630 . . . . . . . 8 (𝑦 = π‘Œ β†’ ((𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) ↔ (𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (π‘ŒπΌπ‘§))))
32 oveq2 7417 . . . . . . . . . . 11 (𝑦 = π‘Œ β†’ (𝑒𝐼𝑦) = (π‘’πΌπ‘Œ))
3332eleq2d 2820 . . . . . . . . . 10 (𝑦 = π‘Œ β†’ (π‘Ž ∈ (𝑒𝐼𝑦) ↔ π‘Ž ∈ (π‘’πΌπ‘Œ)))
3433anbi1d 631 . . . . . . . . 9 (𝑦 = π‘Œ β†’ ((π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼𝑋)) ↔ (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋))))
3534rexbidv 3179 . . . . . . . 8 (𝑦 = π‘Œ β†’ (βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼𝑋)) ↔ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋))))
3631, 35imbi12d 345 . . . . . . 7 (𝑦 = π‘Œ β†’ (((𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼𝑋))) ↔ ((𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (π‘ŒπΌπ‘§)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋)))))
37362ralbidv 3219 . . . . . 6 (𝑦 = π‘Œ β†’ (βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼𝑋))) ↔ βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (π‘ŒπΌπ‘§)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋)))))
38 oveq2 7417 . . . . . . . . . 10 (𝑧 = 𝑍 β†’ (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
3938eleq2d 2820 . . . . . . . . 9 (𝑧 = 𝑍 β†’ (𝑒 ∈ (𝑋𝐼𝑧) ↔ 𝑒 ∈ (𝑋𝐼𝑍)))
40 oveq2 7417 . . . . . . . . . 10 (𝑧 = 𝑍 β†’ (π‘ŒπΌπ‘§) = (π‘ŒπΌπ‘))
4140eleq2d 2820 . . . . . . . . 9 (𝑧 = 𝑍 β†’ (𝑣 ∈ (π‘ŒπΌπ‘§) ↔ 𝑣 ∈ (π‘ŒπΌπ‘)))
4239, 41anbi12d 632 . . . . . . . 8 (𝑧 = 𝑍 β†’ ((𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (π‘ŒπΌπ‘§)) ↔ (𝑒 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘))))
4342imbi1d 342 . . . . . . 7 (𝑧 = 𝑍 β†’ (((𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (π‘ŒπΌπ‘§)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋))) ↔ ((𝑒 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋)))))
44432ralbidv 3219 . . . . . 6 (𝑧 = 𝑍 β†’ (βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (π‘ŒπΌπ‘§)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋))) ↔ βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋)))))
4528, 37, 44rspc3v 3628 . . . . 5 ((𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) β†’ (βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (π‘₯𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼π‘₯))) β†’ βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋)))))
4617, 18, 19, 45syl3anc 1372 . . . 4 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (π‘₯𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼π‘₯))) β†’ βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋)))))
4716, 46mpd 15 . . 3 (πœ‘ β†’ βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋))))
48 axtgpasch.4 . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑃)
49 axtgpasch.5 . . . 4 (πœ‘ β†’ 𝑉 ∈ 𝑃)
50 eleq1 2822 . . . . . . 7 (𝑒 = π‘ˆ β†’ (𝑒 ∈ (𝑋𝐼𝑍) ↔ π‘ˆ ∈ (𝑋𝐼𝑍)))
5150anbi1d 631 . . . . . 6 (𝑒 = π‘ˆ β†’ ((𝑒 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘)) ↔ (π‘ˆ ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘))))
52 oveq1 7416 . . . . . . . . 9 (𝑒 = π‘ˆ β†’ (π‘’πΌπ‘Œ) = (π‘ˆπΌπ‘Œ))
5352eleq2d 2820 . . . . . . . 8 (𝑒 = π‘ˆ β†’ (π‘Ž ∈ (π‘’πΌπ‘Œ) ↔ π‘Ž ∈ (π‘ˆπΌπ‘Œ)))
5453anbi1d 631 . . . . . . 7 (𝑒 = π‘ˆ β†’ ((π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋)) ↔ (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋))))
5554rexbidv 3179 . . . . . 6 (𝑒 = π‘ˆ β†’ (βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋)) ↔ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋))))
5651, 55imbi12d 345 . . . . 5 (𝑒 = π‘ˆ β†’ (((𝑒 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋))) ↔ ((π‘ˆ ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋)))))
57 eleq1 2822 . . . . . . 7 (𝑣 = 𝑉 β†’ (𝑣 ∈ (π‘ŒπΌπ‘) ↔ 𝑉 ∈ (π‘ŒπΌπ‘)))
5857anbi2d 630 . . . . . 6 (𝑣 = 𝑉 β†’ ((π‘ˆ ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘)) ↔ (π‘ˆ ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (π‘ŒπΌπ‘))))
59 oveq1 7416 . . . . . . . . 9 (𝑣 = 𝑉 β†’ (𝑣𝐼𝑋) = (𝑉𝐼𝑋))
6059eleq2d 2820 . . . . . . . 8 (𝑣 = 𝑉 β†’ (π‘Ž ∈ (𝑣𝐼𝑋) ↔ π‘Ž ∈ (𝑉𝐼𝑋)))
6160anbi2d 630 . . . . . . 7 (𝑣 = 𝑉 β†’ ((π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋)) ↔ (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑉𝐼𝑋))))
6261rexbidv 3179 . . . . . 6 (𝑣 = 𝑉 β†’ (βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋)) ↔ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑉𝐼𝑋))))
6358, 62imbi12d 345 . . . . 5 (𝑣 = 𝑉 β†’ (((π‘ˆ ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋))) ↔ ((π‘ˆ ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (π‘ŒπΌπ‘)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑉𝐼𝑋)))))
6456, 63rspc2v 3623 . . . 4 ((π‘ˆ ∈ 𝑃 ∧ 𝑉 ∈ 𝑃) β†’ (βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋))) β†’ ((π‘ˆ ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (π‘ŒπΌπ‘)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑉𝐼𝑋)))))
6548, 49, 64syl2anc 585 . . 3 (πœ‘ β†’ (βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((𝑒 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋))) β†’ ((π‘ˆ ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (π‘ŒπΌπ‘)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑉𝐼𝑋)))))
6647, 65mpd 15 . 2 (πœ‘ β†’ ((π‘ˆ ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (π‘ŒπΌπ‘)) β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑉𝐼𝑋))))
671, 2, 66mp2and 698 1 (πœ‘ β†’ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑉𝐼𝑋)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∨ w3o 1087   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433  Vcvv 3475  [wsbc 3778   βˆ– cdif 3946   ∩ cin 3948  π’« cpw 4603  {csn 4629  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  Basecbs 17144  distcds 17206  TarskiGcstrkg 27678  TarskiGCcstrkgc 27679  TarskiGBcstrkgb 27680  TarskiGCBcstrkgcb 27681  Itvcitv 27684  LineGclng 27685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-trkgb 27700  df-trkg 27704
This theorem is referenced by:  tgbtwncom  27739  tgbtwnswapid  27743  tgbtwnintr  27744  tgtrisegint  27750  tgbtwnconn1  27826  midexlem  27943  opphllem  27986  opphllem1  27998  outpasch  28006  hlpasch  28007  lnopp2hpgb  28014  f1otrg  28122
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