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Theorem frege120 42329
Description: Simplified application of one direction of dffrege115 42324. 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 42267 . . 3 (∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋) → [𝐴 / 𝑎](𝑌𝑅𝑎𝑎 = 𝑋))
3 sbcim1 3800 . . . 4 ([𝐴 / 𝑎](𝑌𝑅𝑎𝑎 = 𝑋) → ([𝐴 / 𝑎]𝑌𝑅𝑎[𝐴 / 𝑎]𝑎 = 𝑋))
4 sbcbr2g 5168 . . . . . 6 (𝐴𝑊 → ([𝐴 / 𝑎]𝑌𝑅𝑎𝑌𝑅𝐴 / 𝑎𝑎))
51, 4ax-mp 5 . . . . 5 ([𝐴 / 𝑎]𝑌𝑅𝑎𝑌𝑅𝐴 / 𝑎𝑎)
6 csbvarg 4396 . . . . . . 7 (𝐴𝑊𝐴 / 𝑎𝑎 = 𝐴)
71, 6ax-mp 5 . . . . . 6 𝐴 / 𝑎𝑎 = 𝐴
87breq2i 5118 . . . . 5 (𝑌𝑅𝐴 / 𝑎𝑎𝑌𝑅𝐴)
95, 8bitri 275 . . . 4 ([𝐴 / 𝑎]𝑌𝑅𝑎𝑌𝑅𝐴)
10 sbceq1g 4379 . . . . . 6 (𝐴𝑊 → ([𝐴 / 𝑎]𝑎 = 𝑋𝐴 / 𝑎𝑎 = 𝑋))
111, 10ax-mp 5 . . . . 5 ([𝐴 / 𝑎]𝑎 = 𝑋𝐴 / 𝑎𝑎 = 𝑋)
127eqeq1i 2742 . . . . 5 (𝐴 / 𝑎𝑎 = 𝑋𝐴 = 𝑋)
1311, 12bitri 275 . . . 4 ([𝐴 / 𝑎]𝑎 = 𝑋𝐴 = 𝑋)
143, 9, 133imtr3g 295 . . 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 42328 . 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 1540   = wceq 1542  wcel 2107  [wsbc 3744  csb 3860   class class class wbr 5110  ccnv 5637  Fun wfun 6495
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-frege1 42136  ax-frege2 42137  ax-frege8 42155  ax-frege52a 42203  ax-frege58b 42247
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-fun 6503
This theorem is referenced by:  frege121  42330
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