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Theorem fveqsb 38178
Description: Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
Hypotheses
Ref Expression
fveqsb.2 (𝑥 = (𝐹𝐴) → (𝜑𝜓))
fveqsb.3 𝑥𝜓
Assertion
Ref Expression
fveqsb (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐹,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem fveqsb
StepHypRef Expression
1 fvex 6168 . . 3 (𝐹𝐴) ∈ V
2 fveqsb.3 . . . 4 𝑥𝜓
3 fveqsb.2 . . . 4 (𝑥 = (𝐹𝐴) → (𝜑𝜓))
42, 3sbciegf 3454 . . 3 ((𝐹𝐴) ∈ V → ([(𝐹𝐴) / 𝑥]𝜑𝜓))
51, 4ax-mp 5 . 2 ([(𝐹𝐴) / 𝑥]𝜑𝜓)
6 fvsb 38177 . 2 (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
75, 6syl5bbr 274 1 (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥) ∧ 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wnf 1705  wcel 1987  ∃!weu 2469  Vcvv 3190  [wsbc 3422   class class class wbr 4623  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-sn 4156  df-pr 4158  df-uni 4410  df-iota 5820  df-fv 5865
This theorem is referenced by: (None)
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