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Theorem sbccom2f 33101
Description: Commutative law for double class substitution, with non free 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 3425 . . . 4 ([𝐵 / 𝑧][𝑧 / 𝑦]𝜑[𝐵 / 𝑦]𝜑)
21bicomi 213 . . 3 ([𝐵 / 𝑦]𝜑[𝐵 / 𝑧][𝑧 / 𝑦]𝜑)
32sbcbii 3458 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑧][𝑧 / 𝑦]𝜑)
4 sbccom2f.1 . . 3 𝐴 ∈ V
54sbccom2 33100 . 2 ([𝐴 / 𝑥][𝐵 / 𝑧][𝑧 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
6 vex 3176 . . . . . . 7 𝑧 ∈ V
76sbccom2 33100 . . . . . 6 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝑧 / 𝑦𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
8 sbccom2f.2 . . . . . . . 8 𝑦𝐴
9 eqidd 2611 . . . . . . . 8 (𝑦 = 𝑧𝐴 = 𝐴)
106, 8, 9csbief 3524 . . . . . . 7 𝑧 / 𝑦𝐴 = 𝐴
11 dfsbcq 3404 . . . . . . 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 263 . . . . 5 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
1413bicomi 213 . . . 4 ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
1514sbcbii 3458 . . 3 ([𝐴 / 𝑥𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑧][𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
16 sbcco 3425 . . 3 ([𝐴 / 𝑥𝐵 / 𝑧][𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
1715, 16bitri 263 . 2 ([𝐴 / 𝑥𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
183, 5, 173bitri 285 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 195   = wceq 1475  wcel 1977  wnfc 2738  Vcvv 3173  [wsbc 3402  csb 3499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500
This theorem is referenced by:  sbccom2fi  33102
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