Theorem a16g 1875

Description: A generalization of Axiom ax-16 1825. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
a16g	|- ( A. x x = y -> ( ph -> A. z ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem a16g

Step	Hyp	Ref	Expression
1		aev 1823	. 2 |- ( A. x x = y -> A. z z = x )
2		ax16 1824	. 2 |- ( A. x x = y -> ( ph -> A. x ph ) )
3		biidd 172	. . . 4 |- ( A. z z = x -> ( ph <-> ph ) )
4	3	dral1 1741	. . 3 |- ( A. z z = x -> ( A. z ph <-> A. x ph ) )
5	4	biimprd 158	. 2 |- ( A. z z = x -> ( A. x ph -> A. z ph ) )
6	1, 2, 5	sylsyld 58	1 |- ( A. x x = y -> ( ph -> A. z ph ) )

Syntax hints:   -> wi 4   A.wal 1362

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  a16gb  1876  a16nf  1877