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Theorem sbcco3g 3976
Description: Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
Hypothesis
Ref Expression
sbcco3g.1 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
sbcco3g (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcco3g
StepHypRef Expression
1 sbcnestg 3974 . 2 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
2 elex 3201 . . 3 (𝐴𝑉𝐴 ∈ V)
3 nfcvd 2762 . . . 4 (𝐴 ∈ V → 𝑥𝐶)
4 sbcco3g.1 . . . 4 (𝑥 = 𝐴𝐵 = 𝐶)
53, 4csbiegf 3542 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐶)
6 dfsbcq 3423 . . 3 (𝐴 / 𝑥𝐵 = 𝐶 → ([𝐴 / 𝑥𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
72, 5, 63syl 18 . 2 (𝐴𝑉 → ([𝐴 / 𝑥𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
81, 7bitrd 268 1 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  Vcvv 3189  [wsbc 3421  csb 3518
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
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-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-sbc 3422  df-csb 3519
This theorem is referenced by:  fzshftral  12377  2rexfrabdioph  36875  3rexfrabdioph  36876  4rexfrabdioph  36877  6rexfrabdioph  36878  7rexfrabdioph  36879
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