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Theorem frege118 39057
Description: Simplified application of one direction of dffrege115 39054. 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 38997 . . 3 (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) → [𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
3 sbcimg 3675 . . . . 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 4900 . . . . . . 7 (𝑌𝑉 → ([𝑌 / 𝑏]𝑏𝑅𝑋𝑌 / 𝑏𝑏𝑅𝑋))
61, 5ax-mp 5 . . . . . 6 ([𝑌 / 𝑏]𝑏𝑅𝑋𝑌 / 𝑏𝑏𝑅𝑋)
7 csbvarg 4198 . . . . . . . 8 (𝑌𝑉𝑌 / 𝑏𝑏 = 𝑌)
81, 7ax-mp 5 . . . . . . 7 𝑌 / 𝑏𝑏 = 𝑌
98breq1i 4850 . . . . . 6 (𝑌 / 𝑏𝑏𝑅𝑋𝑌𝑅𝑋)
106, 9bitri 267 . . . . 5 ([𝑌 / 𝑏]𝑏𝑅𝑋𝑌𝑅𝑋)
11 sbcal 3683 . . . . . 6 ([𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋) ↔ ∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋))
12 sbcimg 3675 . . . . . . . . 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 4900 . . . . . . . . . . 11 (𝑌𝑉 → ([𝑌 / 𝑏]𝑏𝑅𝑎𝑌 / 𝑏𝑏𝑅𝑎))
151, 14ax-mp 5 . . . . . . . . . 10 ([𝑌 / 𝑏]𝑏𝑅𝑎𝑌 / 𝑏𝑏𝑅𝑎)
168breq1i 4850 . . . . . . . . . 10 (𝑌 / 𝑏𝑏𝑅𝑎𝑌𝑅𝑎)
1715, 16bitri 267 . . . . . . . . 9 ([𝑌 / 𝑏]𝑏𝑅𝑎𝑌𝑅𝑎)
18 sbcg 3699 . . . . . . . . . 10 (𝑌𝑉 → ([𝑌 / 𝑏]𝑎 = 𝑋𝑎 = 𝑋))
191, 18ax-mp 5 . . . . . . . . 9 ([𝑌 / 𝑏]𝑎 = 𝑋𝑎 = 𝑋)
2017, 19imbi12i 342 . . . . . . . 8 (([𝑌 / 𝑏]𝑏𝑅𝑎[𝑌 / 𝑏]𝑎 = 𝑋) ↔ (𝑌𝑅𝑎𝑎 = 𝑋))
2113, 20bitri 267 . . . . . . 7 ([𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋) ↔ (𝑌𝑅𝑎𝑎 = 𝑋))
2221albii 1915 . . . . . 6 (∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎𝑎 = 𝑋) ↔ ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))
2311, 22bitri 267 . . . . 5 ([𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋) ↔ ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))
2410, 23imbi12i 342 . . . 4 (([𝑌 / 𝑏]𝑏𝑅𝑋[𝑌 / 𝑏]𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) ↔ (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
254, 24bitri 267 . . 3 ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) ↔ (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
262, 25sylib 210 . 2 (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))
27 frege116.x . . 3 𝑋𝑈
2827frege117 39056 . 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 198  wal 1651   = wceq 1653  wcel 2157  [wsbc 3633  csb 3728   class class class wbr 4843  ccnv 5311  Fun wfun 6095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-frege1 38866  ax-frege2 38867  ax-frege8 38885  ax-frege52a 38933  ax-frege58b 38977
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ifp 1087  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-fun 6103
This theorem is referenced by:  frege119  39058
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