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Theorem frege116 38590
Description: One direction of dffrege115 38589. Proposition 116 of [Frege1879] p. 77. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege116.x 𝑋𝑈
Assertion
Ref Expression
frege116 (Fun 𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
Distinct variable groups:   𝑎,𝑏,𝑅   𝑋,𝑎,𝑏
Allowed substitution hints:   𝑈(𝑎,𝑏)

Proof of Theorem frege116
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 dffrege115 38589 . 2 (∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅)
2 frege116.x . . . 4 𝑋𝑈
32frege68c 38542 . . 3 ((∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅) → (Fun 𝑅[𝑋 / 𝑐]𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐))))
4 sbcal 3518 . . . 4 ([𝑋 / 𝑐]𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
5 sbcimg 3510 . . . . . . 7 (𝑋𝑈 → ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑐[𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐))))
62, 5ax-mp 5 . . . . . 6 ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑐[𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐)))
7 sbcbr2g 4743 . . . . . . . . 9 (𝑋𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑐𝑏𝑅𝑋 / 𝑐𝑐))
82, 7ax-mp 5 . . . . . . . 8 ([𝑋 / 𝑐]𝑏𝑅𝑐𝑏𝑅𝑋 / 𝑐𝑐)
9 csbvarg 4036 . . . . . . . . . 10 (𝑋𝑈𝑋 / 𝑐𝑐 = 𝑋)
102, 9ax-mp 5 . . . . . . . . 9 𝑋 / 𝑐𝑐 = 𝑋
1110breq2i 4693 . . . . . . . 8 (𝑏𝑅𝑋 / 𝑐𝑐𝑏𝑅𝑋)
128, 11bitri 264 . . . . . . 7 ([𝑋 / 𝑐]𝑏𝑅𝑐𝑏𝑅𝑋)
13 sbcal 3518 . . . . . . . 8 ([𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐) ↔ ∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐))
14 sbcimg 3510 . . . . . . . . . . 11 (𝑋𝑈 → ([𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑎[𝑋 / 𝑐]𝑎 = 𝑐)))
152, 14ax-mp 5 . . . . . . . . . 10 ([𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑎[𝑋 / 𝑐]𝑎 = 𝑐))
16 sbcg 3536 . . . . . . . . . . . 12 (𝑋𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑎𝑏𝑅𝑎))
172, 16ax-mp 5 . . . . . . . . . . 11 ([𝑋 / 𝑐]𝑏𝑅𝑎𝑏𝑅𝑎)
18 sbceq2g 4023 . . . . . . . . . . . . 13 (𝑋𝑈 → ([𝑋 / 𝑐]𝑎 = 𝑐𝑎 = 𝑋 / 𝑐𝑐))
192, 18ax-mp 5 . . . . . . . . . . . 12 ([𝑋 / 𝑐]𝑎 = 𝑐𝑎 = 𝑋 / 𝑐𝑐)
2010eqeq2i 2663 . . . . . . . . . . . 12 (𝑎 = 𝑋 / 𝑐𝑐𝑎 = 𝑋)
2119, 20bitri 264 . . . . . . . . . . 11 ([𝑋 / 𝑐]𝑎 = 𝑐𝑎 = 𝑋)
2217, 21imbi12i 339 . . . . . . . . . 10 (([𝑋 / 𝑐]𝑏𝑅𝑎[𝑋 / 𝑐]𝑎 = 𝑐) ↔ (𝑏𝑅𝑎𝑎 = 𝑋))
2315, 22bitri 264 . . . . . . . . 9 ([𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ (𝑏𝑅𝑎𝑎 = 𝑋))
2423albii 1787 . . . . . . . 8 (∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋))
2513, 24bitri 264 . . . . . . 7 ([𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋))
2612, 25imbi12i 339 . . . . . 6 (([𝑋 / 𝑐]𝑏𝑅𝑐[𝑋 / 𝑐]𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
276, 26bitri 264 . . . . 5 ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
2827albii 1787 . . . 4 (∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
294, 28bitri 264 . . 3 ([𝑋 / 𝑐]𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
303, 29syl6ib 241 . 2 ((∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅) → (Fun 𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋))))
311, 30ax-mp 5 1 (Fun 𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521   = wceq 1523  wcel 2030  [wsbc 3468  csb 3566   class class class wbr 4685  ccnv 5142  Fun wfun 5920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-frege1 38401  ax-frege2 38402  ax-frege8 38420  ax-frege52a 38468  ax-frege58b 38512
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-fun 5928
This theorem is referenced by:  frege117  38591
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