Theorem sbcco2 2786

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 2768	. 2 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝑥 / 𝑦][𝐵 / 𝑥]𝜑)
2		nfv 1421	. . 3 ⊢ Ⅎ𝑦[𝐴 / 𝑥]𝜑
3		sbcco2.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
4	3	equcoms 1594	. . . 4 ⊢ (𝑦 = 𝑥 → 𝐴 = 𝐵)
5		dfsbcq 2766	. . . . 5 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑))
6	5	bicomd 129	. . . 4 ⊢ (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
7	4, 6	syl 14	. . 3 ⊢ (𝑦 = 𝑥 → ([𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
8	2, 7	sbie 1674	. 2 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)
9	1, 8	bitr3i 175	1 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243  [wsb 1645  [wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-sbc 2765

This theorem is referenced by: (None)