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Theorem sbcco3gw 4383
Description: Composition of two substitutions. Version of sbcco3g 4388 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 27-Nov-2005.) Avoid ax-13 2371. (Revised by Gino Giotto, 26-Jan-2024.)
Hypothesis
Ref Expression
sbcco3gw.1 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
sbcco3gw (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝐶   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcco3gw
StepHypRef Expression
1 sbcnestgw 4381 . 2 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
2 elex 3462 . . 3 (𝐴𝑉𝐴 ∈ V)
3 nfcvd 2905 . . . 4 (𝐴 ∈ V → 𝑥𝐶)
4 sbcco3gw.1 . . . 4 (𝑥 = 𝐴𝐵 = 𝐶)
53, 4csbiegf 3890 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐶)
6 dfsbcq 3742 . . 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 279 1 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  Vcvv 3444  [wsbc 3740  csb 3856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3446  df-sbc 3741  df-csb 3857
This theorem is referenced by:  fzshftral  13535  2rexfrabdioph  41162  3rexfrabdioph  41163  4rexfrabdioph  41164  6rexfrabdioph  41165  7rexfrabdioph  41166
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