Theorem cbvsbcw 3821

Description: Change bound variables in a wff substitution. Version of cbvsbc 3823 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Jeff Hankins, 19-Sep-2009.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.)

Hypotheses

Ref	Expression
cbvsbcw.1	⊢ Ⅎ𝑦𝜑
cbvsbcw.2	⊢ Ⅎ𝑥𝜓
cbvsbcw.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvsbcw	⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbvsbcw

Step	Hyp	Ref	Expression
1		cbvsbcw.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2		cbvsbcw.2	. . . 4 ⊢ Ⅎ𝑥𝜓
3		cbvsbcw.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvabw 2813	. . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
5	4	eleq2i 2833	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑦 ∣ 𝜓})
6		df-sbc 3789	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
7		df-sbc 3789	. 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓})
8	5, 6, 7	3bitr4i 303	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)

Syntax hints:   → wi 4   ↔ wb 206  Ⅎwnf 1783   ∈ wcel 2108  {cab 2714  [wsbc 3788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-sbc 3789

This theorem is referenced by:  cbvcsbw  3909