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	⊢ Ⅎ𝑦𝐴
sbccom2fi.3	⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
sbccom2fi.4	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)

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

Step	Hyp	Ref	Expression
1		sbccom2fi.1	. . 3 ⊢ 𝐴 ∈ V
2		sbccom2fi.2	. . 3 ⊢ Ⅎ𝑦𝐴
3	1, 2	sbccom2f 34244	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
4		sbccom2fi.3	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
5		dfsbcq 3578	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → ([⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))
6	4, 5	ax-mp 5	. 2 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)
7		sbccom2fi.4	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)
8	7	sbcbii 3632	. 2 ⊢ ([𝐶 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐶 / 𝑦]𝜓)
9	3, 6, 8	3bitri 286	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜓)

Syntax hints:   ↔ wb 196   = wceq 1632   ∈ wcel 2139  Ⅎwnfc 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