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 )