Theorem sbccom2fi 36285

Description: Commutative law for double class substitution, with nonfree 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 36284	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
4		sbccom2fi.3	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
5		dfsbcq 3718	. . 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 3776	. 2 ⊢ ([𝐶 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐶 / 𝑦]𝜓)
9	3, 6, 8	3bitri 297	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜓)

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2106  Ⅎwnfc 2887  Vcvv 3432  [wsbc 3716  ⦋csb 3832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-v 3434  df-sbc 3717  df-csb 3833

This theorem is referenced by:  csbcom2fi  36286