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Metamath Proof Explorer

Theorem pm11.53

Description: Theorem *11.53 in WhiteheadRussell p. 164. See pm11.53v for a version requiring fewer axioms. (Contributed by Andrew Salmon, 24-May-2011)

Ref Expression
Assertion pm11.53 
|- ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) )

Proof

Step Hyp Ref Expression
1 19.21v 
 |-  ( A. y ( ph -> ps ) <-> ( ph -> A. y ps ) )
2 1 albii 
 |-  ( A. x A. y ( ph -> ps ) <-> A. x ( ph -> A. y ps ) )
3 nfv 
 |-  F/ x ps
4 3 nfal 
 |-  F/ x A. y ps
5 4 19.23 
 |-  ( A. x ( ph -> A. y ps ) <-> ( E. x ph -> A. y ps ) )
6 2 5 bitri 
 |-  ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) )