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Theorem sbccom2 34416
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 3656 . . . . . . 7 ([𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐵 / 𝑦]𝜑)
21bicomi 216 . . . . . 6 ([𝐵 / 𝑦]𝜑[𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
32sbcbii 3689 . . . . 5 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
4 sbcco 3656 . . . . . 6 ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
54bicomi 216 . . . . 5 ([𝐴 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑)
6 vex 3388 . . . . . . 7 𝑧 ∈ V
76sbccom2lem 34415 . . . . . 6 ([𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
87sbcbii 3689 . . . . 5 ([𝐴 / 𝑧][𝑧 / 𝑥][𝐵 / 𝑤][𝑤 / 𝑦]𝜑[𝐴 / 𝑧][𝑧 / 𝑥𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
93, 5, 83bitri 289 . . . 4 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑧][𝑧 / 𝑥𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
10 sbccom2.1 . . . . 5 𝐴 ∈ V
1110sbccom2lem 34415 . . . 4 ([𝐴 / 𝑧][𝑧 / 𝑥𝐵 / 𝑤][𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝐴 / 𝑧𝑧 / 𝑥𝐵 / 𝑤][𝐴 / 𝑧][𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
12 sbcco 3656 . . . . 5 ([𝐴 / 𝑧][𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
1312sbcbii 3689 . . . 4 ([𝐴 / 𝑧𝑧 / 𝑥𝐵 / 𝑤][𝐴 / 𝑧][𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝐴 / 𝑧𝑧 / 𝑥𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
149, 11, 133bitri 289 . . 3 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑧𝑧 / 𝑥𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
15 csbco 3738 . . . 4 𝐴 / 𝑧𝑧 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
16 dfsbcq 3635 . . . 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 267 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑)
19 sbccom 3705 . . 3 ([𝐴 / 𝑥][𝑤 / 𝑦]𝜑[𝑤 / 𝑦][𝐴 / 𝑥]𝜑)
2019sbcbii 3689 . 2 ([𝐴 / 𝑥𝐵 / 𝑤][𝐴 / 𝑥][𝑤 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑥]𝜑)
21 sbcco 3656 . 2 ([𝐴 / 𝑥𝐵 / 𝑤][𝑤 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
2218, 20, 213bitri 289 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1653  wcel 2157  Vcvv 3385  [wsbc 3633  csb 3728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-sbc 3634  df-csb 3729
This theorem is referenced by:  sbccom2f  34417
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