Theorem sbcco2 3747

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 3724	. 2 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝑥 / 𝑦][𝐵 / 𝑥]𝜑)
2		sbcco2.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
3	2	equcoms 2027	. . . 4 ⊢ (𝑦 = 𝑥 → 𝐴 = 𝐵)
4		dfsbcq 3722	. . . . 5 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑))
5	4	bicomd 226	. . . 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 2100	. 2 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)
8	1, 7	bitr3i 280	1 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  [wsb 2069  [wsbc 3720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-sbc 3721

This theorem is referenced by:  tfinds2  7558