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Metamath Proof Explorer

Theorem 19.41rg

Description: Closed form of right-to-left implication of 19.41 , Theorem 19.41 of Margaris p. 90. Derived from 19.41rgVD . (Contributed by Alan Sare, 8-Feb-2014) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion 19.41rg 
|- ( A. x ( ps -> A. x ps ) -> ( ( E. x ph /\ ps ) -> E. x ( ph /\ ps ) ) )

Proof

Step Hyp Ref Expression
1 sp 
 |-  ( A. x ( ps -> A. x ps ) -> ( ps -> A. x ps ) )
2 pm3.21 
 |-  ( ps -> ( ph -> ( ph /\ ps ) ) )
3 2 a1i 
 |-  ( ( ps -> A. x ps ) -> ( ps -> ( ph -> ( ph /\ ps ) ) ) )
4 3 al2imi 
 |-  ( A. x ( ps -> A. x ps ) -> ( A. x ps -> A. x ( ph -> ( ph /\ ps ) ) ) )
5 exim 
 |-  ( A. x ( ph -> ( ph /\ ps ) ) -> ( E. x ph -> E. x ( ph /\ ps ) ) )
6 4 5 syl6 
 |-  ( A. x ( ps -> A. x ps ) -> ( A. x ps -> ( E. x ph -> E. x ( ph /\ ps ) ) ) )
7 1 6 syld 
 |-  ( A. x ( ps -> A. x ps ) -> ( ps -> ( E. x ph -> E. x ( ph /\ ps ) ) ) )
8 7 com23 
 |-  ( A. x ( ps -> A. x ps ) -> ( E. x ph -> ( ps -> E. x ( ph /\ ps ) ) ) )
9 8 impd 
 |-  ( A. x ( ps -> A. x ps ) -> ( ( E. x ph /\ ps ) -> E. x ( ph /\ ps ) ) )