Theorem sb6f

Description: Equivalence for substitution when y is not free in ph . The implication "to the left" is sb2 and does not require the nonfreeness hypothesis. Theorem sb6 replaces the nonfreeness hypothesis with a disjoint variable condition on x , y and requires fewer axioms. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Jun-1993) (Revised by Mario Carneiro, 4-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sb6f.1	|- F/ y ph
Assertion	sb6f	|- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )

Proof

Step	Hyp	Ref	Expression
1		sb6f.1	|- F/ y ph
2	1	nf5ri	|- ( ph -> A. y ph )
3	2	sbimi	|- ( [ y / x ] ph -> [ y / x ] A. y ph )
4		sb4a	|- ( [ y / x ] A. y ph -> A. x ( x = y -> ph ) )
5	3 4	syl	|- ( [ y / x ] ph -> A. x ( x = y -> ph ) )
6		sb2	|- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
7	5 6	impbii	|- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )