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Theorem sbccom2f 38113
Description: Commutative law for double class substitution, with nonfree variable condition. (Contributed by Giovanni Mascellani, 31-May-2019.)
Hypotheses
Ref Expression
sbccom2f.1 𝐴 ∈ V
sbccom2f.2 𝑦𝐴
Assertion
Ref Expression
sbccom2f ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem sbccom2f
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sbccow 3814 . . . 4 ([𝐵 / 𝑧][𝑧 / 𝑦]𝜑[𝐵 / 𝑦]𝜑)
21bicomi 224 . . 3 ([𝐵 / 𝑦]𝜑[𝐵 / 𝑧][𝑧 / 𝑦]𝜑)
32sbcbii 3852 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑧][𝑧 / 𝑦]𝜑)
4 sbccom2f.1 . . 3 𝐴 ∈ V
54sbccom2 38112 . 2 ([𝐴 / 𝑥][𝐵 / 𝑧][𝑧 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
6 vex 3482 . . . . . . 7 𝑧 ∈ V
76sbccom2 38112 . . . . . 6 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝑧 / 𝑦𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
8 sbccom2f.2 . . . . . . . 8 𝑦𝐴
96, 8csbgfi 3929 . . . . . . 7 𝑧 / 𝑦𝐴 = 𝐴
10 dfsbcq 3793 . . . . . . 7 (𝑧 / 𝑦𝐴 = 𝐴 → ([𝑧 / 𝑦𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥][𝑧 / 𝑦]𝜑))
119, 10ax-mp 5 . . . . . 6 ([𝑧 / 𝑦𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
127, 11bitri 275 . . . . 5 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
1312bicomi 224 . . . 4 ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
1413sbcbii 3852 . . 3 ([𝐴 / 𝑥𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑧][𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
15 sbccow 3814 . . 3 ([𝐴 / 𝑥𝐵 / 𝑧][𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
1614, 15bitri 275 . 2 ([𝐴 / 𝑥𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
173, 5, 163bitri 297 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wcel 2106  wnfc 2888  Vcvv 3478  [wsbc 3791  csb 3908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480  df-sbc 3792  df-csb 3909
This theorem is referenced by:  sbccom2fi  38114
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