Theorem sb6a 2007

Description: Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
sb6a	|- ( [ y / x ] ph <-> A. x ( x = y -> [ x / y ] ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem sb6a

Step	Hyp	Ref	Expression
1		sb6 1901	. 2 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
2		sbequ12 1785	. . . . 5 |- ( y = x -> ( ph <-> [ x / y ] ph ) )
3	2	equcoms 1722	. . . 4 |- ( x = y -> ( ph <-> [ x / y ] ph ) )
4	3	pm5.74i 180	. . 3 |- ( ( x = y -> ph ) <-> ( x = y -> [ x / y ] ph ) )
5	4	albii 1484	. 2 |- ( A. x ( x = y -> ph ) <-> A. x ( x = y -> [ x / y ] ph ) )
6	1, 5	bitri 184	1 |- ( [ y / x ] ph <-> A. x ( x = y -> [ x / y ] ph ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   [wsb 1776

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-sb 1777

This theorem is referenced by: (None)