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Theorem sbcco3g 3061
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 3057 . 2 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
2 elex 2700 . . 3 (𝐴𝑉𝐴 ∈ V)
3 nfcvd 2283 . . . 4 (𝐴 ∈ V → 𝑥𝐶)
4 sbcco3g.1 . . . 4 (𝑥 = 𝐴𝐵 = 𝐶)
53, 4csbiegf 3047 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐶)
6 dfsbcq 2914 . . 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 187 1 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1332  wcel 1481  Vcvv 2689  [wsbc 2912  csb 3006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2913  df-csb 3007
This theorem is referenced by:  fzshftral  9918
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