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Theorem sbcco3gw 4323
 Description: Composition of two substitutions. Version of sbcco3g 4328 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 27-Nov-2005.) (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 4321 . 2 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
2 elex 3429 . . 3 (𝐴𝑉𝐴 ∈ V)
3 nfcvd 2921 . . . 4 (𝐴 ∈ V → 𝑥𝐶)
4 sbcco3gw.1 . . . 4 (𝑥 = 𝐴𝐵 = 𝐶)
53, 4csbiegf 3841 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐶)
6 dfsbcq 3701 . . 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 282 1 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1539   ∈ wcel 2112  Vcvv 3410  [wsbc 3699  ⦋csb 3808 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-v 3412  df-sbc 3700  df-csb 3809 This theorem is referenced by:  fzshftral  13058  2rexfrabdioph  40156  3rexfrabdioph  40157  4rexfrabdioph  40158  6rexfrabdioph  40159  7rexfrabdioph  40160
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