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Metamath Proof Explorer

Theorem 19.12vvv

Description: Version of 19.12vv with a disjoint variable condition, requiring fewer axioms. See also 19.12 . (Contributed by BJ, 18-Mar-2020)

Ref Expression
Assertion 19.12vvv 
|- ( E. x A. y ( ph -> ps ) <-> A. y E. x ( ph -> ps ) )

Proof

Step Hyp Ref Expression
1 19.21v 
 |-  ( A. y ( ph -> ps ) <-> ( ph -> A. y ps ) )
2 1 exbii 
 |-  ( E. x A. y ( ph -> ps ) <-> E. x ( ph -> A. y ps ) )
3 19.36v 
 |-  ( E. x ( ph -> A. y ps ) <-> ( A. x ph -> A. y ps ) )
4 19.36v 
 |-  ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) )
5 4 albii 
 |-  ( A. y E. x ( ph -> ps ) <-> A. y ( A. x ph -> ps ) )
6 19.21v 
 |-  ( A. y ( A. x ph -> ps ) <-> ( A. x ph -> A. y ps ) )
7 5 6 bitr2i 
 |-  ( ( A. x ph -> A. y ps ) <-> A. y E. x ( ph -> ps ) )
8 2 3 7 3bitri 
 |-  ( E. x A. y ( ph -> ps ) <-> A. y E. x ( ph -> ps ) )