Theorem hbnaes 1723

Description: Rule that applies hbnae 1721 to antecedent. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
hbnalequs.1	|- ( A. z -. A. x x = y -> ph )

Assertion

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

Proof of Theorem hbnaes

Step	Hyp	Ref	Expression
1		hbnae 1721	. 2 |- ( -. A. x x = y -> A. z -. A. x x = y )
2		hbnalequs.1	. 2 |- ( A. z -. A. x x = y -> ph )
3	1, 2	syl 14	1 |- ( -. A. x x = y -> ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1351

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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359

This theorem is referenced by:  sbal2  2020