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Metamath Proof Explorer

Theorem 19.12vv

Description: Special case of 19.12 where its converse holds. See 19.12vvv for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 18-Jul-2001) (Revised by Andrew Salmon, 11-Jul-2011)

Ref Expression
Assertion 19.12vv 
|- ( 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 nfv 
 |-  F/ x ps
4 3 nfal 
 |-  F/ x A. y ps
5 4 19.36 
 |-  ( E. x ( ph -> A. y ps ) <-> ( A. x ph -> A. y ps ) )
6 19.36v 
 |-  ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) )
7 6 albii 
 |-  ( A. y E. x ( ph -> ps ) <-> A. y ( A. x ph -> ps ) )
8 nfv 
 |-  F/ y ph
9 8 nfal 
 |-  F/ y A. x ph
10 9 19.21 
 |-  ( A. y ( A. x ph -> ps ) <-> ( A. x ph -> A. y ps ) )
11 7 10 bitr2i 
 |-  ( ( A. x ph -> A. y ps ) <-> A. y E. x ( ph -> ps ) )
12 2 5 11 3bitri 
 |-  ( E. x A. y ( ph -> ps ) <-> A. y E. x ( ph -> ps ) )