Theorem sbccom2fi 39152

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 39151	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
4		sbccom2fi.3	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
5		dfsbcq 3740	. . 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 3794	. 2 ⊢ ([𝐶 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐶 / 𝑦]𝜓)
9	3, 6, 8	3bitri 300	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜓)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2144  Ⅎwnfc 2907  Vcvv 3450  [wsbc 3738  ⦋csb 3846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-v 3452  df-sbc 3739  df-csb 3847

This theorem is used by:  csbcom2fi  39153