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Metamath Proof Explorer

Theorem 2sb6rf

Description: Reversed double substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 3-Feb-2005) (Revised by Mario Carneiro, 6-Oct-2016) Remove variable constraints. (Revised by Wolf Lammen, 28-Sep-2018) (Proof shortened by Wolf Lammen, 13-Apr-2023) (New usage is discouraged.)

Ref Expression
Hypotheses 2sb5rf.1 
|- F/ z ph
2sb5rf.2 
|- F/ w ph
Assertion 2sb6rf 
|- ( ph <-> A. z A. w ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) )

Proof

Step Hyp Ref Expression
1 2sb5rf.1 
 |-  F/ z ph
2 2sb5rf.2 
 |-  F/ w ph
3 1 19.23 
 |-  ( A. z ( E. w ( z = x /\ w = y ) -> ph ) <-> ( E. z E. w ( z = x /\ w = y ) -> ph ) )
4 2 19.23 
 |-  ( A. w ( ( z = x /\ w = y ) -> ph ) <-> ( E. w ( z = x /\ w = y ) -> ph ) )
5 4 albii 
 |-  ( A. z A. w ( ( z = x /\ w = y ) -> ph ) <-> A. z ( E. w ( z = x /\ w = y ) -> ph ) )
6 2ax6e 
 |-  E. z E. w ( z = x /\ w = y )
7 6 a1bi 
 |-  ( ph <-> ( E. z E. w ( z = x /\ w = y ) -> ph ) )
8 3 5 7 3bitr4ri 
 |-  ( ph <-> A. z A. w ( ( z = x /\ w = y ) -> ph ) )
9 sbequ12r 
 |-  ( z = x -> ( [ z / x ] [ w / y ] ph <-> [ w / y ] ph ) )
10 sbequ12r 
 |-  ( w = y -> ( [ w / y ] ph <-> ph ) )
11 9 10 sylan9bb 
 |-  ( ( z = x /\ w = y ) -> ( [ z / x ] [ w / y ] ph <-> ph ) )
12 11 pm5.74i 
 |-  ( ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) <-> ( ( z = x /\ w = y ) -> ph ) )
13 12 2albii 
 |-  ( A. z A. w ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) <-> A. z A. w ( ( z = x /\ w = y ) -> ph ) )
14 8 13 bitr4i 
 |-  ( ph <-> A. z A. w ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) )