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Theorem frege118 38775
Description: Simplified application of one direction of dffrege115 38772. Proposition 118 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege116.x 𝑋𝑈
frege118.y 𝑌𝑉
Assertion
Ref Expression
frege118 (Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
Distinct variable groups:   𝑅,𝑎   𝑋,𝑎   𝑌,𝑎
Allowed substitution hints:   𝑈(𝑎)   𝑉(𝑎)

Proof of Theorem frege118
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 frege118.y . . . 4 𝑌𝑉
21frege58c 38715 . . 3 (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) → [𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
3 sbcimg 3616 . . . . 5 (𝑌𝑉 → ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑋[𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋))))
41, 3ax-mp 5 . . . 4 ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑋[𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
5 sbcbr1g 4859 . . . . . . 7 (𝑌𝑉 → ([𝑌 / 𝑏]𝑏𝑅𝑋𝑌 / 𝑏𝑏𝑅𝑋))
61, 5ax-mp 5 . . . . . 6 ([𝑌 / 𝑏]𝑏𝑅𝑋𝑌 / 𝑏𝑏𝑅𝑋)
7 csbvarg 4144 . . . . . . . 8 (𝑌𝑉𝑌 / 𝑏𝑏 = 𝑌)
81, 7ax-mp 5 . . . . . . 7 𝑌 / 𝑏𝑏 = 𝑌
98breq1i 4809 . . . . . 6 (𝑌 / 𝑏𝑏𝑅𝑋𝑌𝑅𝑋)
106, 9bitri 264 . . . . 5 ([𝑌 / 𝑏]𝑏𝑅𝑋𝑌𝑅𝑋)
11 sbcal 3624 . . . . . 6 ([𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋) ↔ ∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋))
12 sbcimg 3616 . . . . . . . . 9 (𝑌𝑉 → ([𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑎[𝑌 / 𝑏]𝑎 = 𝑋)))
131, 12ax-mp 5 . . . . . . . 8 ([𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑎[𝑌 / 𝑏]𝑎 = 𝑋))
14 sbcbr1g 4859 . . . . . . . . . . 11 (𝑌𝑉 → ([𝑌 / 𝑏]𝑏𝑅𝑎𝑌 / 𝑏𝑏𝑅𝑎))
151, 14ax-mp 5 . . . . . . . . . 10 ([𝑌 / 𝑏]𝑏𝑅𝑎𝑌 / 𝑏𝑏𝑅𝑎)
168breq1i 4809 . . . . . . . . . 10 (𝑌 / 𝑏𝑏𝑅𝑎𝑌𝑅𝑎)
1715, 16bitri 264 . . . . . . . . 9 ([𝑌 / 𝑏]𝑏𝑅𝑎𝑌𝑅𝑎)
18 sbcg 3642 . . . . . . . . . 10 (𝑌𝑉 → ([𝑌 / 𝑏]𝑎 = 𝑋𝑎 = 𝑋))
191, 18ax-mp 5 . . . . . . . . 9 ([𝑌 / 𝑏]𝑎 = 𝑋𝑎 = 𝑋)
2017, 19imbi12i 339 . . . . . . . 8 (([𝑌 / 𝑏]𝑏𝑅𝑎[𝑌 / 𝑏]𝑎 = 𝑋) ↔ (𝑌𝑅𝑎𝑎 = 𝑋))
2113, 20bitri 264 . . . . . . 7 ([𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋) ↔ (𝑌𝑅𝑎𝑎 = 𝑋))
2221albii 1894 . . . . . 6 (∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋) ↔ ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))
2311, 22bitri 264 . . . . 5 ([𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋) ↔ ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))
2410, 23imbi12i 339 . . . 4 (([𝑌 / 𝑏]𝑏𝑅𝑋[𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) ↔ (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
254, 24bitri 264 . . 3 ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) ↔ (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
262, 25sylib 208 . 2 (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
27 frege116.x . . 3 𝑋𝑈
2827frege117 38774 . 2 ((∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))) → (Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))))
2926, 28ax-mp 5 1 (Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1628   = wceq 1630  wcel 2137  [wsbc 3574  csb 3672   class class class wbr 4802  ccnv 5263  Fun wfun 6041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053  ax-frege1 38584  ax-frege2 38585  ax-frege8 38603  ax-frege52a 38651  ax-frege58b 38695
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1051  df-3an 1074  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-br 4803  df-opab 4863  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-fun 6049
This theorem is referenced by:  frege119  38776
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