Theorem cbvsbcw 3059

Description: Version of cbvsbc 3060 with a disjoint variable condition. (Contributed 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 2354	. . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
5	4	eleq2i 2298	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑦 ∣ 𝜓})
6		df-sbc 3032	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
7		df-sbc 3032	. 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓})
8	5, 6, 7	3bitr4i 212	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1508   ∈ wcel 2202  {cab 2217  [wsbc 3031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-sbc 3032

This theorem is referenced by:  cbvcsbw  3131