This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem sb8v

Description: Substitution of variable in universal quantifier. Version of sb8f with a disjoint variable condition replacing the nonfree hypothesis F/ y ph , not requiring ax-12 . (Contributed by SN, 5-Dec-2024)

Ref Expression
Assertion sb8v 
|- ( A. x ph <-> A. y [ y / x ] ph )

Proof

Step Hyp Ref Expression
1 sb6 
 |-  ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
2 1 albii 
 |-  ( A. y [ y / x ] ph <-> A. y A. x ( x = y -> ph ) )
3 alcom 
 |-  ( A. y A. x ( x = y -> ph ) <-> A. x A. y ( x = y -> ph ) )
4 equcom 
 |-  ( x = y <-> y = x )
5 4 imbi1i 
 |-  ( ( x = y -> ph ) <-> ( y = x -> ph ) )
6 5 albii 
 |-  ( A. y ( x = y -> ph ) <-> A. y ( y = x -> ph ) )
7 equsv 
 |-  ( A. y ( y = x -> ph ) <-> ph )
8 6 7 bitri 
 |-  ( A. y ( x = y -> ph ) <-> ph )
9 8 albii 
 |-  ( A. x A. y ( x = y -> ph ) <-> A. x ph )
10 2 3 9 3bitrri 
 |-  ( A. x ph <-> A. y [ y / x ] ph )