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Theorem frege118 43308
Description: Simplified application of one direction of dffrege115 43305. 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 43248 . . 3 (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) → [𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
3 sbcimg 3823 . . . . 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 5198 . . . . . . 7 (𝑌𝑉 → ([𝑌 / 𝑏]𝑏𝑅𝑋𝑌 / 𝑏𝑏𝑅𝑋))
61, 5ax-mp 5 . . . . . 6 ([𝑌 / 𝑏]𝑏𝑅𝑋𝑌 / 𝑏𝑏𝑅𝑋)
7 csbvarg 4426 . . . . . . . 8 (𝑌𝑉𝑌 / 𝑏𝑏 = 𝑌)
81, 7ax-mp 5 . . . . . . 7 𝑌 / 𝑏𝑏 = 𝑌
98breq1i 5148 . . . . . 6 (𝑌 / 𝑏𝑏𝑅𝑋𝑌𝑅𝑋)
106, 9bitri 275 . . . . 5 ([𝑌 / 𝑏]𝑏𝑅𝑋𝑌𝑅𝑋)
11 sbcal 3836 . . . . . 6 ([𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋) ↔ ∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋))
12 sbcimg 3823 . . . . . . . . 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 5198 . . . . . . . . . . 11 (𝑌𝑉 → ([𝑌 / 𝑏]𝑏𝑅𝑎𝑌 / 𝑏𝑏𝑅𝑎))
151, 14ax-mp 5 . . . . . . . . . 10 ([𝑌 / 𝑏]𝑏𝑅𝑎𝑌 / 𝑏𝑏𝑅𝑎)
168breq1i 5148 . . . . . . . . . 10 (𝑌 / 𝑏𝑏𝑅𝑎𝑌𝑅𝑎)
1715, 16bitri 275 . . . . . . . . 9 ([𝑌 / 𝑏]𝑏𝑅𝑎𝑌𝑅𝑎)
18 sbcg 3851 . . . . . . . . . 10 (𝑌𝑉 → ([𝑌 / 𝑏]𝑎 = 𝑋𝑎 = 𝑋))
191, 18ax-mp 5 . . . . . . . . 9 ([𝑌 / 𝑏]𝑎 = 𝑋𝑎 = 𝑋)
2017, 19imbi12i 350 . . . . . . . 8 (([𝑌 / 𝑏]𝑏𝑅𝑎[𝑌 / 𝑏]𝑎 = 𝑋) ↔ (𝑌𝑅𝑎𝑎 = 𝑋))
2113, 20bitri 275 . . . . . . 7 ([𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋) ↔ (𝑌𝑅𝑎𝑎 = 𝑋))
2221albii 1813 . . . . . 6 (∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋) ↔ ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))
2311, 22bitri 275 . . . . 5 ([𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋) ↔ ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))
2410, 23imbi12i 350 . . . 4 (([𝑌 / 𝑏]𝑏𝑅𝑋[𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) ↔ (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
254, 24bitri 275 . . 3 ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) ↔ (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
262, 25sylib 217 . 2 (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
27 frege116.x . . 3 𝑋𝑈
2827frege117 43307 . 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 205  wal 1531   = wceq 1533  wcel 2098  [wsbc 3772  csb 3888   class class class wbr 5141  ccnv 5668  Fun wfun 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-frege1 43117  ax-frege2 43118  ax-frege8 43136  ax-frege52a 43184  ax-frege58b 43228
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1060  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-fun 6539
This theorem is referenced by:  frege119  43309
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