Theorem sbcco2 3752

Description: A composition law for class substitution. Importantly, 𝑥 may occur free in the class expression substituted for 𝐴. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypothesis

Ref	Expression
sbcco2.1	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
sbcco2	⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem sbcco2

Step	Hyp	Ref	Expression
1		sbsbc 3729	. 2 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝑥 / 𝑦][𝐵 / 𝑥]𝜑)
2		sbcco2.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
3	2	equcoms 2022	. . . 4 ⊢ (𝑦 = 𝑥 → 𝐴 = 𝐵)
4		dfsbcq 3727	. . . . 5 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑))
5	4	bicomd 222	. . . 4 ⊢ (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
6	3, 5	syl 17	. . 3 ⊢ (𝑦 = 𝑥 → ([𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
7	6	sbievw 2094	. 2 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)
8	1, 7	bitr3i 276	1 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1540  [wsb 2066  [wsbc 3725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-sbc 3726

This theorem is referenced by:  tfinds2  7753