Theorem sb6a 2017

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 1911	. 2 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
2		sbequ12 1795	. . . . 5 |- ( y = x -> ( ph <-> [ x / y ] ph ) )
3	2	equcoms 1732	. . . 4 |- ( x = y -> ( ph <-> [ x / y ] ph ) )
4	3	pm5.74i 180	. . 3 |- ( ( x = y -> ph ) <-> ( x = y -> [ x / y ] ph ) )
5	4	albii 1494	. 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 1371   [wsb 1786

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-sb 1787

This theorem is referenced by: (None)