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Metamath Proof Explorer

Theorem sb8f

Description: Substitution of variable in universal quantifier. Version of sb8 with a disjoint variable condition, not requiring ax-10 or ax-13 . (Contributed by NM, 16-May-1993) (Revised by Wolf Lammen, 19-Jan-2023) Avoid ax-10 . (Revised by SN, 5-Dec-2024)

Ref Expression
Hypothesis sb8f.nf 
|- F/ y ph
Assertion sb8f 
|- ( A. x ph <-> A. y [ y / x ] ph )

Proof

Step Hyp Ref Expression
1 sb8f.nf 
 |-  F/ y ph
2 sb6 
 |-  ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
3 2 albii 
 |-  ( A. y [ y / x ] ph <-> A. y A. x ( x = y -> ph ) )
4 alcom 
 |-  ( A. y A. x ( x = y -> ph ) <-> A. x A. y ( x = y -> ph ) )
5 sb6 
 |-  ( [ x / y ] ph <-> A. y ( y = x -> ph ) )
6 1 sbf 
 |-  ( [ x / y ] ph <-> ph )
7 equcom 
 |-  ( y = x <-> x = y )
8 7 imbi1i 
 |-  ( ( y = x -> ph ) <-> ( x = y -> ph ) )
9 8 albii 
 |-  ( A. y ( y = x -> ph ) <-> A. y ( x = y -> ph ) )
10 5 6 9 3bitr3ri 
 |-  ( A. y ( x = y -> ph ) <-> ph )
11 10 albii 
 |-  ( A. x A. y ( x = y -> ph ) <-> A. x ph )
12 3 4 11 3bitrri 
 |-  ( A. x ph <-> A. y [ y / x ] ph )