Theorem cbvsbc 3074

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	|- F/yph
cbvsbc.2	|- F/xps
cbvsbc.3	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
cbvsbc	|- ( [.A / x ].ph <-> [.A / y ].ps)

Proof of Theorem cbvsbc

Step	Hyp	Ref	Expression
1		cbvsbc.1	. . . 4 |- F/yph
2		cbvsbc.2	. . . 4 |- F/xps
3		cbvsbc.3	. . . 4 |- (x = y -> (ph <-> ps))
4	1, 2, 3	cbvab 2471	. . 3 |- {x | ph} = {y | ps}
5	4	eleq2i 2417	. 2 |- (A e. {x | ph} <-> A e. {y | ps})
6		df-sbc 3047	. 2 |- ( [.A / x ].ph <-> A e. {x | ph})
7		df-sbc 3047	. 2 |- ( [.A / y ].ps <-> A e. {y | ps})
8	5, 6, 7	3bitr4i 268	1 |- ( [.A / x ].ph <-> [.A / y ].ps)

Syntax hints:   -> wi 4   <-> wb 176   F/wnf 1544   e. wcel 1710  {cab 2339   [.wsbc 3046

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 3047

This theorem is referenced by:  cbvsbcv  3075  cbvcsb  3140