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Theorem sbccom2fi 34245
Description: Commutative law for double class substitution, with non free variable condition and in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.)
Hypotheses
Ref Expression
sbccom2fi.1 𝐴 ∈ V
sbccom2fi.2 𝑦𝐴
sbccom2fi.3 𝐴 / 𝑥𝐵 = 𝐶
sbccom2fi.4 ([𝐴 / 𝑥]𝜑𝜓)
Assertion
Ref Expression
sbccom2fi ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem sbccom2fi
StepHypRef Expression
1 sbccom2fi.1 . . 3 𝐴 ∈ V
2 sbccom2fi.2 . . 3 𝑦𝐴
31, 2sbccom2f 34244 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
4 sbccom2fi.3 . . 3 𝐴 / 𝑥𝐵 = 𝐶
5 dfsbcq 3578 . . 3 (𝐴 / 𝑥𝐵 = 𝐶 → ([𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑))
64, 5ax-mp 5 . 2 ([𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)
7 sbccom2fi.4 . . 3 ([𝐴 / 𝑥]𝜑𝜓)
87sbcbii 3632 . 2 ([𝐶 / 𝑦][𝐴 / 𝑥]𝜑[𝐶 / 𝑦]𝜓)
93, 6, 83bitri 286 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1632  wcel 2139  wnfc 2889  Vcvv 3340  [wsbc 3576  csb 3674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-sbc 3577  df-csb 3675
This theorem is referenced by:  csbcom2fi  34247
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