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Metamath Proof Explorer

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 )