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Theorem sbccom2f 34417
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 sbcco 3656 . . . 4 ([𝐵 / 𝑧][𝑧 / 𝑦]𝜑[𝐵 / 𝑦]𝜑)
21bicomi 216 . . 3 ([𝐵 / 𝑦]𝜑[𝐵 / 𝑧][𝑧 / 𝑦]𝜑)
32sbcbii 3689 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑧][𝑧 / 𝑦]𝜑)
4 sbccom2f.1 . . 3 𝐴 ∈ V
54sbccom2 34416 . 2 ([𝐴 / 𝑥][𝐵 / 𝑧][𝑧 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
6 vex 3388 . . . . . . 7 𝑧 ∈ V
76sbccom2 34416 . . . . . 6 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝑧 / 𝑦𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
8 sbccom2f.2 . . . . . . . 8 𝑦𝐴
9 eqidd 2800 . . . . . . . 8 (𝑦 = 𝑧𝐴 = 𝐴)
106, 8, 9csbief 3753 . . . . . . 7 𝑧 / 𝑦𝐴 = 𝐴
11 dfsbcq 3635 . . . . . . 7 (𝑧 / 𝑦𝐴 = 𝐴 → ([𝑧 / 𝑦𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥][𝑧 / 𝑦]𝜑))
1210, 11ax-mp 5 . . . . . 6 ([𝑧 / 𝑦𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
137, 12bitri 267 . . . . 5 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
1413bicomi 216 . . . 4 ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
1514sbcbii 3689 . . 3 ([𝐴 / 𝑥𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑧][𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
16 sbcco 3656 . . 3 ([𝐴 / 𝑥𝐵 / 𝑧][𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
1715, 16bitri 267 . 2 ([𝐴 / 𝑥𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
183, 5, 173bitri 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  wnfc 2928  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-3an 1110  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:  sbccom2fi  34418
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