Theorem sbccom2fi 38087

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 38086	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
4		sbccom2fi.3	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
5		dfsbcq 3806	. . 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 3865	. 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 206   = wceq 1537   ∈ wcel 2108  Ⅎwnfc 2893  Vcvv 3488  [wsbc 3804  ⦋csb 3921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-v 3490  df-sbc 3805  df-csb 3922

This theorem is referenced by:  csbcom2fi  38088