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Metamath Proof Explorer

Theorem hbsbwOLD

Description: Obsolete version of hbsbw as of 14-May-2025. (Contributed by NM, 12-Aug-1993) (Revised by GG, 23-May-2024) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Hypothesis hbsbw.1 
|- ( ph -> A. z ph )
Assertion hbsbwOLD 
|- ( [ y / x ] ph -> A. z [ y / x ] ph )

Proof

Step Hyp Ref Expression
1 hbsbw.1 
 |-  ( ph -> A. z ph )
2 dfsb 
 |-  ( [ y / x ] ph <-> A. w ( w = y -> A. x ( x = w -> ph ) ) )
3 1 imim2i 
 |-  ( ( x = w -> ph ) -> ( x = w -> A. z ph ) )
4 3 alimi 
 |-  ( A. x ( x = w -> ph ) -> A. x ( x = w -> A. z ph ) )
5 19.21v 
 |-  ( A. z ( x = w -> ph ) <-> ( x = w -> A. z ph ) )
6 5 biimpri 
 |-  ( ( x = w -> A. z ph ) -> A. z ( x = w -> ph ) )
7 6 alimi 
 |-  ( A. x ( x = w -> A. z ph ) -> A. x A. z ( x = w -> ph ) )
8 ax-11 
 |-  ( A. x A. z ( x = w -> ph ) -> A. z A. x ( x = w -> ph ) )
9 4 7 8 3syl 
 |-  ( A. x ( x = w -> ph ) -> A. z A. x ( x = w -> ph ) )
10 9 imim2i 
 |-  ( ( w = y -> A. x ( x = w -> ph ) ) -> ( w = y -> A. z A. x ( x = w -> ph ) ) )
11 19.21v 
 |-  ( A. z ( w = y -> A. x ( x = w -> ph ) ) <-> ( w = y -> A. z A. x ( x = w -> ph ) ) )
12 10 11 sylibr 
 |-  ( ( w = y -> A. x ( x = w -> ph ) ) -> A. z ( w = y -> A. x ( x = w -> ph ) ) )
13 12 hbal 
 |-  ( A. w ( w = y -> A. x ( x = w -> ph ) ) -> A. z A. w ( w = y -> A. x ( x = w -> ph ) ) )
14 2 13 hbxfrbi 
 |-  ( [ y / x ] ph -> A. z [ y / x ] ph )