Theorem nfald2 2176

Description: Variation on nfald 2045 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 2163	. . . . 5 |- F/ y -. A. x x = y
3	1, 2	nfan 2022	. . . 4 |- F/ y ( ph /\ -. A. x x = y )
4		nfald2.2	. . . 4 |- ( ( ph /\ -. A. x x = y ) -> F/ x ps )
5	3, 4	nfald 2045	. . 3 |- ( ( ph /\ -. A. x x = y ) -> F/ x A. y ps )
6	5	ex 440	. 2 |- ( ph -> ( -. A. x x = y -> F/ x A. y ps ) )
7		nfa1 1990	. . 3 |- F/ y A. y ps
8		biidd 245	. . . 4 |- ( A. x x = y -> ( A. y ps <-> A. y ps ) )
9	8	drnf1 2174	. . 3 |- ( A. x x = y -> ( F/ x A. y ps <-> F/ y A. y ps ) )
10	7, 9	mpbiri 241	. 2 |- ( A. x x = y -> F/ x A. y ps )
11	6, 10	pm2.61d2 165	1 |- ( ph -> F/ x A. y ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 375   A.wal 1453   F/wnf 1678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679

This theorem is referenced by:  nfexd2  2177  dvelimf  2179  nfeud2  2322  nfrald  2785  nfiotad  5568  nfixp  7567