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Metamath Proof Explorer

Theorem ax4fromc4

Description: Rederivation of Axiom ax-4 from ax-c4 , ax-c5 , ax-gen and minimal implicational calculus { ax-mp , ax-1 , ax-2 }. See axc4 for the derivation of ax-c4 from ax-4 . (Contributed by NM, 23-May-2008) (Proof modification is discouraged.) Use ax-4 instead. (New usage is discouraged.)

Ref Expression
Assertion ax4fromc4 
|- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )

Proof

Step Hyp Ref Expression
1 ax-c4 
 |-  ( A. x ( A. x ( ph -> ps ) -> ( A. x ph -> ps ) ) -> ( A. x ( ph -> ps ) -> A. x ( A. x ph -> ps ) ) )
2 ax-c5 
 |-  ( A. x ph -> ph )
3 ax-c5 
 |-  ( A. x ( ph -> ps ) -> ( ph -> ps ) )
4 2 3 syl5 
 |-  ( A. x ( ph -> ps ) -> ( A. x ph -> ps ) )
5 1 4 mpg 
 |-  ( A. x ( ph -> ps ) -> A. x ( A. x ph -> ps ) )
6 ax-c4 
 |-  ( A. x ( A. x ph -> ps ) -> ( A. x ph -> A. x ps ) )
7 5 6 syl 
 |-  ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )