Theorem nfeqd

Description: Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016)

	Ref	Expression
Hypotheses	nfeqd.1	|- ( ph -> F/_ x A )
	nfeqd.2	|- ( ph -> F/_ x B )
Assertion	nfeqd	|- ( ph -> F/ x A = B )

Proof

Step	Hyp	Ref	Expression
1		nfeqd.1	|- ( ph -> F/_ x A )
2		nfeqd.2	|- ( ph -> F/_ x B )
3		dfcleq	|- ( A = B <-> A. y ( y e. A <-> y e. B ) )
4		nfv	|- F/ y ph
5		df-nfc	|- ( F/_ x A <-> A. y F/ x y e. A )
6	1 5	sylib	|- ( ph -> A. y F/ x y e. A )
7	6	19.21bi	|- ( ph -> F/ x y e. A )
8		df-nfc	|- ( F/_ x B <-> A. y F/ x y e. B )
9	2 8	sylib	|- ( ph -> A. y F/ x y e. B )
10	9	19.21bi	|- ( ph -> F/ x y e. B )
11	7 10	nfbid	|- ( ph -> F/ x ( y e. A <-> y e. B ) )
12	4 11	nfald	|- ( ph -> F/ x A. y ( y e. A <-> y e. B ) )
13	3 12	nfxfrd	|- ( ph -> F/ x A = B )