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Theorem sbccom2fi 35292
Description: Commutative law for double class substitution, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.)
Hypotheses
Ref Expression
sbccom2fi.1 𝐴 ∈ V
sbccom2fi.2 𝑦𝐴
sbccom2fi.3 𝐴 / 𝑥𝐵 = 𝐶
sbccom2fi.4 ([𝐴 / 𝑥]𝜑𝜓)
Assertion
Ref Expression
sbccom2fi ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem sbccom2fi
StepHypRef Expression
1 sbccom2fi.1 . . 3 𝐴 ∈ V
2 sbccom2fi.2 . . 3 𝑦𝐴
31, 2sbccom2f 35291 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
4 sbccom2fi.3 . . 3 𝐴 / 𝑥𝐵 = 𝐶
5 dfsbcq 3778 . . 3 (𝐴 / 𝑥𝐵 = 𝐶 → ([𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑))
64, 5ax-mp 5 . 2 ([𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)
7 sbccom2fi.4 . . 3 ([𝐴 / 𝑥]𝜑𝜓)
87sbcbii 3833 . 2 ([𝐶 / 𝑦][𝐴 / 𝑥]𝜑[𝐶 / 𝑦]𝜓)
93, 6, 83bitri 298 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1530  wcel 2107  wnfc 2966  Vcvv 3500  [wsbc 3776  csb 3887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-v 3502  df-sbc 3777  df-csb 3888
This theorem is referenced by:  csbcom2fi  35293
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