Theorem vtocl2OLD

Description: Obsolete proof of vtocl2 as of 23-Aug-2023. (Contributed by NM, 26-Jul-1995) (Proof shortened by Andrew Salmon, 8-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	vtocl2.1	|- A e. _V
	vtocl2.2	|- B e. _V
	vtocl2.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
	vtocl2.4	|- ph
Assertion	vtocl2OLD	|- ps

Proof

Step	Hyp	Ref	Expression
1		vtocl2.1	|- A e. _V
2		vtocl2.2	|- B e. _V
3		vtocl2.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
4		vtocl2.4	|- ph
5	1	isseti	|- E. x x = A
6	2	isseti	|- E. y y = B
7		exdistrv	|- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
8	3	biimpd	|- ( ( x = A /\ y = B ) -> ( ph -> ps ) )
9	8	2eximi	|- ( E. x E. y ( x = A /\ y = B ) -> E. x E. y ( ph -> ps ) )
10	7 9	sylbir	|- ( ( E. x x = A /\ E. y y = B ) -> E. x E. y ( ph -> ps ) )
11	5 6 10	mp2an	|- E. x E. y ( ph -> ps )
12		19.36v	|- ( E. y ( ph -> ps ) <-> ( A. y ph -> ps ) )
13	12	exbii	|- ( E. x E. y ( ph -> ps ) <-> E. x ( A. y ph -> ps ) )
14	11 13	mpbi	|- E. x ( A. y ph -> ps )
15	14	19.36iv	|- ( A. x A. y ph -> ps )
16	4	ax-gen	|- A. y ph
17	15 16	mpg	|- ps