Theorem cbvsbc 3075

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 2472	. . 3 ⊢ {x ∣ φ} = {y ∣ ψ}
5	4	eleq2i 2417	. 2 ⊢ (A ∈ {x ∣ φ} ↔ A ∈ {y ∣ ψ})
6		df-sbc 3048	. 2 ⊢ ([̣A / x]̣φ ↔ A ∈ {x ∣ φ})
7		df-sbc 3048	. 2 ⊢ ([̣A / y]̣ψ ↔ A ∈ {y ∣ ψ})
8	5, 6, 7	3bitr4i 268	1 ⊢ ([̣A / x]̣φ ↔ [̣A / y]̣ψ)

Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544   ∈ wcel 1710  {cab 2339  [̣wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sbc 3048

This theorem is referenced by:  cbvsbcv  3076  cbvcsb  3141