Theorem sb5f1 917

Description: Equivalence for substitution. Similar to Theorem 6.2 of [Quine] p. 40.

Hypothesis

Ref	Expression
sb5f1.1	|- (ph -> A.xph)

Assertion

Ref	Expression
sb5f1	|- (ph <-> A.x(x = y -> [x / y]ph))

Proof of Theorem sb5f1

Step	Hyp	Ref	Expression
1		sb5f1.1	. . 3 |- (ph -> A.xph)
2		sbequ1 863	. . . . 5 |- (y = x -> (ph -> [x / y]ph))
3	2	eqcoms 813	. . . 4 |- (x = y -> (ph -> [x / y]ph))
4	3	com12 13	. . 3 |- (ph -> (x = y -> [x / y]ph))
5	1, 4	19.21ai 740	. 2 |- (ph -> A.x(x = y -> [x / y]ph))
6		sb2 859	. . . 4 |- (A.x(x = y -> [x / y]ph) -> [y / x][x / y]ph)
7		sbco 910	. . . 4 |- ([y / x][x / y]ph <-> [y / x]ph)
8	6, 7	sylib 173	. . 3 |- (A.x(x = y -> [x / y]ph) -> [y / x]ph)
9	1	sbf 870	. . 3 |- ([y / x]ph <-> ph)
10	8, 9	sylib 173	. 2 |- (A.x(x = y -> [x / y]ph) -> ph)
11	5, 10	impbi 139	1 |- (ph <-> A.x(x = y -> [x / y]ph))

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672   = weq 797  [wsb 852

This theorem is referenced by:  eu1 1019

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853

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