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Theorem frege120 43650
Description: Simplified application of one direction of dffrege115 43645. 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 43588 . . 3 (∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋) → [𝐴 / 𝑎](𝑌𝑅𝑎𝑎 = 𝑋))
3 sbcim1 3833 . . . 4 ([𝐴 / 𝑎](𝑌𝑅𝑎𝑎 = 𝑋) → ([𝐴 / 𝑎]𝑌𝑅𝑎[𝐴 / 𝑎]𝑎 = 𝑋))
4 sbcbr2g 5211 . . . . . 6 (𝐴𝑊 → ([𝐴 / 𝑎]𝑌𝑅𝑎𝑌𝑅𝐴 / 𝑎𝑎))
51, 4ax-mp 5 . . . . 5 ([𝐴 / 𝑎]𝑌𝑅𝑎𝑌𝑅𝐴 / 𝑎𝑎)
6 csbvarg 4436 . . . . . . 7 (𝐴𝑊𝐴 / 𝑎𝑎 = 𝐴)
71, 6ax-mp 5 . . . . . 6 𝐴 / 𝑎𝑎 = 𝐴
87breq2i 5161 . . . . 5 (𝑌𝑅𝐴 / 𝑎𝑎𝑌𝑅𝐴)
95, 8bitri 274 . . . 4 ([𝐴 / 𝑎]𝑌𝑅𝑎𝑌𝑅𝐴)
10 sbceq1g 4419 . . . . . 6 (𝐴𝑊 → ([𝐴 / 𝑎]𝑎 = 𝑋𝐴 / 𝑎𝑎 = 𝑋))
111, 10ax-mp 5 . . . . 5 ([𝐴 / 𝑎]𝑎 = 𝑋𝐴 / 𝑎𝑎 = 𝑋)
127eqeq1i 2731 . . . . 5 (𝐴 / 𝑎𝑎 = 𝑋𝐴 = 𝑋)
1311, 12bitri 274 . . . 4 ([𝐴 / 𝑎]𝑎 = 𝑋𝐴 = 𝑋)
143, 9, 133imtr3g 294 . . 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 43649 . 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 205  wal 1532   = wceq 1534  wcel 2099  [wsbc 3776  csb 3892   class class class wbr 5153  ccnv 5681  Fun wfun 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-frege1 43457  ax-frege2 43458  ax-frege8 43476  ax-frege52a 43524  ax-frege58b 43568
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-fun 6556
This theorem is referenced by:  frege121  43651
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