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Theorem sbcco3g 2903
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 2899 . 2 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
2 elex 2566 . . 3 (𝐴𝑉𝐴 ∈ V)
3 nfcvd 2179 . . . 4 (𝐴 ∈ V → 𝑥𝐶)
4 sbcco3g.1 . . . 4 (𝑥 = 𝐴𝐵 = 𝐶)
53, 4csbiegf 2890 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐶)
6 dfsbcq 2766 . . 3 (𝐴 / 𝑥𝐵 = 𝐶 → ([𝐴 / 𝑥𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
72, 5, 63syl 17 . 2 (𝐴𝑉 → ([𝐴 / 𝑥𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
81, 7bitrd 177 1 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  wcel 1393  Vcvv 2557  [wsbc 2764  csb 2852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853
This theorem is referenced by:  fzshftral  8970
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