Theorem cbvsbcw 3056

Description: Version of cbvsbc 3057 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 2352	. . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
5	4	eleq2i 2296	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑦 ∣ 𝜓})
6		df-sbc 3029	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
7		df-sbc 3029	. 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 1506   ∈ wcel 2200  {cab 2215  [wsbc 3028

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-sbc 3029

This theorem is referenced by:  cbvcsbw  3128