Theorem nfald2 2139

Description: Variation on nfald 2011 which adds the hypothesis that x and y are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)

Hypotheses

Ref	Expression
nfald2.1	|- F/ y ph
nfald2.2	|- ( ( ph /\ -. A. x x = y ) -> F/ x ps )

Assertion

Ref	Expression
nfald2	|- ( ph -> F/ x A. y ps )

Proof of Theorem nfald2

Step	Hyp	Ref	Expression
1		nfald2.1	. . . . 5 |- F/ y ph
2		nfnae 2124	. . . . 5 |- F/ y -. A. x x = y
3	1, 2	nfan 1988	. . . 4 |- F/ y ( ph /\ -. A. x x = y )
4		nfald2.2	. . . 4 |- ( ( ph /\ -. A. x x = y ) -> F/ x ps )
5	3, 4	nfald 2011	. . 3 |- ( ( ph /\ -. A. x x = y ) -> F/ x A. y ps )
6	5	ex 435	. 2 |- ( ph -> ( -. A. x x = y -> F/ x A. y ps ) )
7		nfa1 1956	. . 3 |- F/ y A. y ps
8		biidd 240	. . . 4 |- ( A. x x = y -> ( A. y ps <-> A. y ps ) )
9	8	drnf1 2137	. . 3 |- ( A. x x = y -> ( F/ x A. y ps <-> F/ y A. y ps ) )
10	7, 9	mpbiri 236	. 2 |- ( A. x x = y -> F/ x A. y ps )
11	6, 10	pm2.61d2 163	1 |- ( ph -> F/ x A. y ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   A.wal 1435   F/wnf 1661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063

This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662

This theorem is referenced by:  nfexd2  2140  dvelimf  2142  nfeud2  2288  nfrald  2750  nfiotad  5511  nfixp  7496