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Metamath Proof Explorer

Theorem spimvALT

Description: Alternate proof of spimv . Note that it requires only ax-1 through ax-5 together with ax6e . Currently, proofs derive from ax6v , but if ax-6 could be used instead, this proof would reduce axiom usage. (Contributed by NM, 31-Jul-1993) Remove dependency on ax-10 . (Revised by BJ, 29-Nov-2020) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypothesis spimv.1 
|- ( x = y -> ( ph -> ps ) )
Assertion spimvALT 
|- ( A. x ph -> ps )

Proof

Step Hyp Ref Expression
1 spimv.1 
 |-  ( x = y -> ( ph -> ps ) )
2 ax6e 
 |-  E. x x = y
3 2 1 eximii 
 |-  E. x ( ph -> ps )
4 3 19.36iv 
 |-  ( A. x ph -> ps )