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Theorem sbccom2 35407
 Description: Commutative law for double class substitution. (Contributed by Giovanni Mascellani, 31-May-2019.)
Hypothesis
Ref Expression
sbccom2.1 𝐴 ∈ V
Assertion
Ref Expression
sbccom2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem sbccom2
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcco 3801 . . . . . . 7 ([𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐵 / 𝑦]𝜑)
21bicomi 226 . . . . . 6 ([𝐵 / 𝑦]𝜑[𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
32sbcbii 3832 . . . . 5 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
4 sbcco 3801 . . . . . 6 ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
54bicomi 226 . . . . 5 ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
6 vex 3500 . . . . . . 7 𝑧 ∈ V
76sbccom2lem 35406 . . . . . 6 ([𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
87sbcbii 3832 . . . . 5 ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐴 / 𝑧][𝑧 / 𝑥𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
93, 5, 83bitri 299 . . . 4 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑧][𝑧 / 𝑥𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
10 sbccom2.1 . . . . 5 𝐴 ∈ V
1110sbccom2lem 35406 . . . 4 ([𝐴 / 𝑧][𝑧 / 𝑥𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝐴 / 𝑧𝑧 / 𝑥𝐵 / 𝑤][𝐴 / 𝑧][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
12 sbcco 3801 . . . . 5 ([𝐴 / 𝑧][𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
1312sbcbii 3832 . . . 4 ([𝐴 / 𝑧𝑧 / 𝑥𝐵 / 𝑤][𝐴 / 𝑧][𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝐴 / 𝑧𝑧 / 𝑥𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
149, 11, 133bitri 299 . . 3 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑧𝑧 / 𝑥𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
15 csbco 3902 . . . 4 𝐴 / 𝑧𝑧 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
16 dfsbcq 3777 . . . 4 (𝐴 / 𝑧𝑧 / 𝑥𝐵 = 𝐴 / 𝑥𝐵 → ([𝐴 / 𝑧𝑧 / 𝑥𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑))
1715, 16ax-mp 5 . . 3 ([𝐴 / 𝑧𝑧 / 𝑥𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
1814, 17bitri 277 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
19 sbccom 3857 . . 3 ([𝐴 / 𝑥][𝑤 / 𝑦]𝜑[𝑤 / 𝑦][𝐴 / 𝑥]𝜑)
2019sbcbii 3832 . 2 ([𝐴 / 𝑥𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑥]𝜑)
21 sbcco 3801 . 2 ([𝐴 / 𝑥𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
2218, 20, 213bitri 299 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   = wceq 1536   ∈ wcel 2113  Vcvv 3497  [wsbc 3775  ⦋csb 3886 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-13 2389  ax-ext 2796 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-sbc 3776  df-csb 3887 This theorem is referenced by:  sbccom2f  35408
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