Theorem cbvsbc 2479

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

Hypotheses

Ref	Expression
cbvsbc.1	|- (ph -> A.yph)
cbvsbc.2	|- (ps -> A.xps)
cbvsbc.3	|- (x = y -> (ph <-> ps))

Assertion

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

Proof of Theorem cbvsbc

Step	Hyp	Ref	Expression
1		cbvsbc.1	. . . . 5 |- (ph -> A.yph)
2		cbvsbc.2	. . . . 5 |- (ps -> A.xps)
3		cbvsbc.3	. . . . 5 |- (x = y -> (ph <-> ps))
4	1, 2, 3	cbvab 2419	. . . 4 |- {x | ph} = {y | ps}
5	4	eleq2i 1961	. . 3 |- (A e. {x | ph} <-> A e. {y | ps})
6	5	a1i 8	. 2 |- (A e. B -> (A e. {x | ph} <-> A e. {y | ps}))
7		sbc8g 2477	. 2 |- (A e. B -> ([A / x]ph <-> A e. {x | ph}))
8		sbc8g 2477	. 2 |- (A e. B -> ([A / y]ps <-> A e. {y | ps}))
9	6, 7, 8	3bitr4d 609	1 |- (A e. B -> ([A / x]ph <-> [A / y]ps))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  [wsbc 1534  {cab 1871

This theorem is referenced by:  cbvsbcv 2480  ordtypelem6 5689  ordtypelem6OLD 15380

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

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