Theorem equsexALT

Description: Alternate proof of equsex . This proves the result directly, instead of as a corollary of equsal via equs4 . Note in particular that only existential quantifiers appear in the proof and that the only step requiring ax-13 is ax6e . This proof mimics that of equsal (in particular, note that pm5.32i , exbii , 19.41 , mpbiran correspond respectively to pm5.74i , albii , 19.23 , a1bi ). (Contributed by BJ, 20-Aug-2020) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	equsal.1	|- F/ x ps
	equsal.2	|- ( x = y -> ( ph <-> ps ) )
Assertion	equsexALT	|- ( E. x ( x = y /\ ph ) <-> ps )

Proof

Step	Hyp	Ref	Expression
1		equsal.1	|- F/ x ps
2		equsal.2	|- ( x = y -> ( ph <-> ps ) )
3	2	pm5.32i	|- ( ( x = y /\ ph ) <-> ( x = y /\ ps ) )
4	3	exbii	|- ( E. x ( x = y /\ ph ) <-> E. x ( x = y /\ ps ) )
5		ax6e	|- E. x x = y
6	1	19.41	|- ( E. x ( x = y /\ ps ) <-> ( E. x x = y /\ ps ) )
7	5 6	mpbiran	|- ( E. x ( x = y /\ ps ) <-> ps )
8	4 7	bitri	|- ( E. x ( x = y /\ ph ) <-> ps )