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Theorem axtgpasch 27982
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 27968 . . . . . . 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 27970 . . . . . . 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 496 . . . . . 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 1142 . . . . 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 7419 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ (π‘₯𝐼𝑧) = (𝑋𝐼𝑧))
2120eleq2d 2818 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ (𝑒 ∈ (π‘₯𝐼𝑧) ↔ 𝑒 ∈ (𝑋𝐼𝑧)))
2221anbi1d 629 . . . . . . . 8 (π‘₯ = 𝑋 β†’ ((𝑒 ∈ (π‘₯𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) ↔ (𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧))))
23 oveq2 7420 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ (𝑣𝐼π‘₯) = (𝑣𝐼𝑋))
2423eleq2d 2818 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ (π‘Ž ∈ (𝑣𝐼π‘₯) ↔ π‘Ž ∈ (𝑣𝐼𝑋)))
2524anbi2d 628 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ ((π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼π‘₯)) ↔ (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼𝑋))))
2625rexbidv 3177 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼π‘₯)) ↔ βˆƒπ‘Ž ∈ 𝑃 (π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼𝑋))))
2722, 26imbi12d 343 . . . . . . 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 3217 . . . . . 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 7419 . . . . . . . . . 10 (𝑦 = π‘Œ β†’ (𝑦𝐼𝑧) = (π‘ŒπΌπ‘§))
3029eleq2d 2818 . . . . . . . . 9 (𝑦 = π‘Œ β†’ (𝑣 ∈ (𝑦𝐼𝑧) ↔ 𝑣 ∈ (π‘ŒπΌπ‘§)))
3130anbi2d 628 . . . . . . . 8 (𝑦 = π‘Œ β†’ ((𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) ↔ (𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (π‘ŒπΌπ‘§))))
32 oveq2 7420 . . . . . . . . . . 11 (𝑦 = π‘Œ β†’ (𝑒𝐼𝑦) = (π‘’πΌπ‘Œ))
3332eleq2d 2818 . . . . . . . . . 10 (𝑦 = π‘Œ β†’ (π‘Ž ∈ (𝑒𝐼𝑦) ↔ π‘Ž ∈ (π‘’πΌπ‘Œ)))
3433anbi1d 629 . . . . . . . . 9 (𝑦 = π‘Œ β†’ ((π‘Ž ∈ (𝑒𝐼𝑦) ∧ π‘Ž ∈ (𝑣𝐼𝑋)) ↔ (π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋))))
3534rexbidv 3177 . . . . . . . 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 343 . . . . . . 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 3217 . . . . . 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 7420 . . . . . . . . . 10 (𝑧 = 𝑍 β†’ (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
3938eleq2d 2818 . . . . . . . . 9 (𝑧 = 𝑍 β†’ (𝑒 ∈ (𝑋𝐼𝑧) ↔ 𝑒 ∈ (𝑋𝐼𝑍)))
40 oveq2 7420 . . . . . . . . . 10 (𝑧 = 𝑍 β†’ (π‘ŒπΌπ‘§) = (π‘ŒπΌπ‘))
4140eleq2d 2818 . . . . . . . . 9 (𝑧 = 𝑍 β†’ (𝑣 ∈ (π‘ŒπΌπ‘§) ↔ 𝑣 ∈ (π‘ŒπΌπ‘)))
4239, 41anbi12d 630 . . . . . . . 8 (𝑧 = 𝑍 β†’ ((𝑒 ∈ (𝑋𝐼𝑧) ∧ 𝑣 ∈ (π‘ŒπΌπ‘§)) ↔ (𝑒 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘))))
4342imbi1d 340 . . . . . . 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 3217 . . . . . 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 1370 . . . 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 2820 . . . . . . 7 (𝑒 = π‘ˆ β†’ (𝑒 ∈ (𝑋𝐼𝑍) ↔ π‘ˆ ∈ (𝑋𝐼𝑍)))
5150anbi1d 629 . . . . . 6 (𝑒 = π‘ˆ β†’ ((𝑒 ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘)) ↔ (π‘ˆ ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘))))
52 oveq1 7419 . . . . . . . . 9 (𝑒 = π‘ˆ β†’ (π‘’πΌπ‘Œ) = (π‘ˆπΌπ‘Œ))
5352eleq2d 2818 . . . . . . . 8 (𝑒 = π‘ˆ β†’ (π‘Ž ∈ (π‘’πΌπ‘Œ) ↔ π‘Ž ∈ (π‘ˆπΌπ‘Œ)))
5453anbi1d 629 . . . . . . 7 (𝑒 = π‘ˆ β†’ ((π‘Ž ∈ (π‘’πΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋)) ↔ (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋))))
5554rexbidv 3177 . . . . . 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 343 . . . . 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 2820 . . . . . . 7 (𝑣 = 𝑉 β†’ (𝑣 ∈ (π‘ŒπΌπ‘) ↔ 𝑉 ∈ (π‘ŒπΌπ‘)))
5857anbi2d 628 . . . . . 6 (𝑣 = 𝑉 β†’ ((π‘ˆ ∈ (𝑋𝐼𝑍) ∧ 𝑣 ∈ (π‘ŒπΌπ‘)) ↔ (π‘ˆ ∈ (𝑋𝐼𝑍) ∧ 𝑉 ∈ (π‘ŒπΌπ‘))))
59 oveq1 7419 . . . . . . . . 9 (𝑣 = 𝑉 β†’ (𝑣𝐼𝑋) = (𝑉𝐼𝑋))
6059eleq2d 2818 . . . . . . . 8 (𝑣 = 𝑉 β†’ (π‘Ž ∈ (𝑣𝐼𝑋) ↔ π‘Ž ∈ (𝑉𝐼𝑋)))
6160anbi2d 628 . . . . . . 7 (𝑣 = 𝑉 β†’ ((π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑣𝐼𝑋)) ↔ (π‘Ž ∈ (π‘ˆπΌπ‘Œ) ∧ π‘Ž ∈ (𝑉𝐼𝑋))))
6261rexbidv 3177 . . . . . 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 343 . . . . 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 583 . . 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 696 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 395   ∨ w3o 1085   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  {cab 2708  βˆ€wral 3060  βˆƒwrex 3069  {crab 3431  Vcvv 3473  [wsbc 3778   βˆ– cdif 3946   ∩ cin 3948  π’« cpw 4603  {csn 4629  β€˜cfv 6544  (class class class)co 7412   ∈ cmpo 7414  Basecbs 17149  distcds 17211  TarskiGcstrkg 27942  TarskiGCcstrkgc 27943  TarskiGBcstrkgb 27944  TarskiGCBcstrkgcb 27945  Itvcitv 27948  LineGclng 27949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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 7415  df-trkgb 27964  df-trkg 27968
This theorem is referenced by:  tgbtwncom  28003  tgbtwnswapid  28007  tgbtwnintr  28008  tgtrisegint  28014  tgbtwnconn1  28090  midexlem  28207  opphllem  28250  opphllem1  28262  outpasch  28270  hlpasch  28271  lnopp2hpgb  28278  f1otrg  28386
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