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Theorem csbco3g 4361
Description: Composition of two class substitutions. Usage of this theorem is discouraged because it depends on ax-13 2392. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
sbcco3g.1 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
csbco3g (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐷)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝐷(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem csbco3g
StepHypRef Expression
1 csbnestg 4359 . 2 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐴 / 𝑥𝐵 / 𝑦𝐷)
2 elex 3497 . . . 4 (𝐴𝑉𝐴 ∈ V)
3 nfcvd 2981 . . . . 5 (𝐴 ∈ V → 𝑥𝐶)
4 sbcco3g.1 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
53, 4csbiegf 3898 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐶)
62, 5syl 17 . . 3 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐶)
76csbeq1d 3869 . 2 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐷)
81, 7eqtrd 2859 1 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3479  csb 3865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3481  df-sbc 3758  df-csb 3866
This theorem is referenced by: (None)
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