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Metamath Proof Explorer

Theorem pm14.123b

Description: Theorem *14.123 in WhiteheadRussell p. 185. (Contributed by Andrew Salmon, 9-Jun-2011)

Ref Expression
Assertion pm14.123b 
|- ( ( A e. V /\ B e. W ) -> ( ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ [. A / z ]. [. B / w ]. ph ) <-> ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ E. z E. w ph ) ) )

Proof

Step Hyp Ref Expression
1 2sbc5g 
 |-  ( ( A e. V /\ B e. W ) -> ( E. z E. w ( ( z = A /\ w = B ) /\ ph ) <-> [. A / z ]. [. B / w ]. ph ) )
2 1 adantr 
 |-  ( ( ( A e. V /\ B e. W ) /\ A. z A. w ( ph -> ( z = A /\ w = B ) ) ) -> ( E. z E. w ( ( z = A /\ w = B ) /\ ph ) <-> [. A / z ]. [. B / w ]. ph ) )
3 nfa1 
 |-  F/ z A. z A. w ( ph -> ( z = A /\ w = B ) )
4 nfa2 
 |-  F/ w A. z A. w ( ph -> ( z = A /\ w = B ) )
5 simpr 
 |-  ( ( ( z = A /\ w = B ) /\ ph ) -> ph )
6 2sp 
 |-  ( A. z A. w ( ph -> ( z = A /\ w = B ) ) -> ( ph -> ( z = A /\ w = B ) ) )
7 6 ancrd 
 |-  ( A. z A. w ( ph -> ( z = A /\ w = B ) ) -> ( ph -> ( ( z = A /\ w = B ) /\ ph ) ) )
8 5 7 impbid2 
 |-  ( A. z A. w ( ph -> ( z = A /\ w = B ) ) -> ( ( ( z = A /\ w = B ) /\ ph ) <-> ph ) )
9 4 8 exbid 
 |-  ( A. z A. w ( ph -> ( z = A /\ w = B ) ) -> ( E. w ( ( z = A /\ w = B ) /\ ph ) <-> E. w ph ) )
10 3 9 exbid 
 |-  ( A. z A. w ( ph -> ( z = A /\ w = B ) ) -> ( E. z E. w ( ( z = A /\ w = B ) /\ ph ) <-> E. z E. w ph ) )
11 10 adantl 
 |-  ( ( ( A e. V /\ B e. W ) /\ A. z A. w ( ph -> ( z = A /\ w = B ) ) ) -> ( E. z E. w ( ( z = A /\ w = B ) /\ ph ) <-> E. z E. w ph ) )
12 2 11 bitr3d 
 |-  ( ( ( A e. V /\ B e. W ) /\ A. z A. w ( ph -> ( z = A /\ w = B ) ) ) -> ( [. A / z ]. [. B / w ]. ph <-> E. z E. w ph ) )
13 12 pm5.32da 
 |-  ( ( A e. V /\ B e. W ) -> ( ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ [. A / z ]. [. B / w ]. ph ) <-> ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ E. z E. w ph ) ) )