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Theorem frege120 37786
Description: Simplified application of one direction of dffrege115 37781. Proposition 120 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege116.x 𝑋𝑈
frege118.y 𝑌𝑉
frege120.a 𝐴𝑊
Assertion
Ref Expression
frege120 (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝐴 = 𝑋)))

Proof of Theorem frege120
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 frege120.a . . . 4 𝐴𝑊
21frege58c 37724 . . 3 (∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋) → [𝐴 / 𝑎](𝑌𝑅𝑎𝑎 = 𝑋))
3 sbcim1 3468 . . . 4 ([𝐴 / 𝑎](𝑌𝑅𝑎𝑎 = 𝑋) → ([𝐴 / 𝑎]𝑌𝑅𝑎[𝐴 / 𝑎]𝑎 = 𝑋))
4 sbcbr2g 4675 . . . . . 6 (𝐴𝑊 → ([𝐴 / 𝑎]𝑌𝑅𝑎𝑌𝑅𝐴 / 𝑎𝑎))
51, 4ax-mp 5 . . . . 5 ([𝐴 / 𝑎]𝑌𝑅𝑎𝑌𝑅𝐴 / 𝑎𝑎)
6 csbvarg 3980 . . . . . . 7 (𝐴𝑊𝐴 / 𝑎𝑎 = 𝐴)
71, 6ax-mp 5 . . . . . 6 𝐴 / 𝑎𝑎 = 𝐴
87breq2i 4626 . . . . 5 (𝑌𝑅𝐴 / 𝑎𝑎𝑌𝑅𝐴)
95, 8bitri 264 . . . 4 ([𝐴 / 𝑎]𝑌𝑅𝑎𝑌𝑅𝐴)
10 sbceq1g 3965 . . . . . 6 (𝐴𝑊 → ([𝐴 / 𝑎]𝑎 = 𝑋𝐴 / 𝑎𝑎 = 𝑋))
111, 10ax-mp 5 . . . . 5 ([𝐴 / 𝑎]𝑎 = 𝑋𝐴 / 𝑎𝑎 = 𝑋)
127eqeq1i 2626 . . . . 5 (𝐴 / 𝑎𝑎 = 𝑋𝐴 = 𝑋)
1311, 12bitri 264 . . . 4 ([𝐴 / 𝑎]𝑎 = 𝑋𝐴 = 𝑋)
143, 9, 133imtr3g 284 . . 3 ([𝐴 / 𝑎](𝑌𝑅𝑎𝑎 = 𝑋) → (𝑌𝑅𝐴𝐴 = 𝑋))
152, 14syl 17 . 2 (∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋) → (𝑌𝑅𝐴𝐴 = 𝑋))
16 frege116.x . . 3 𝑋𝑈
17 frege118.y . . 3 𝑌𝑉
1816, 17frege119 37785 . 2 ((∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋) → (𝑌𝑅𝐴𝐴 = 𝑋)) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝐴 = 𝑋))))
1915, 18ax-mp 5 1 (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝐴 = 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478   = wceq 1480  wcel 1987  [wsbc 3421  csb 3518   class class class wbr 4618  ccnv 5078  Fun wfun 5846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-frege1 37593  ax-frege2 37594  ax-frege8 37612  ax-frege52a 37660  ax-frege58b 37704
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-fun 5854
This theorem is referenced by:  frege121  37787
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