Theorem cbvsbc 2991

Description: Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

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

Assertion

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

Proof of Theorem cbvsbc

Step	Hyp	Ref	Expression
1		cbvsbc.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2		cbvsbc.2	. . . 4 ⊢ Ⅎ𝑥𝜓
3		cbvsbc.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvab 2301	. . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
5	4	eleq2i 2244	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑦 ∣ 𝜓})
6		df-sbc 2963	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
7		df-sbc 2963	. 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 1460   ∈ wcel 2148  {cab 2163  [wsbc 2962

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-sbc 2963

This theorem is referenced by:  cbvsbcv  2992  cbvcsb  3062