Theorem sb6x

Description: Equivalence involving substitution for a variable not free. Usage of this theorem is discouraged because it depends on ax-13 . Usage of sb6 is preferred, which requires fewer axioms. (Contributed by NM, 2-Jun-1993) (Revised by Mario Carneiro, 4-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sb6x.1	|- F/ x ph
Assertion	sb6x	|- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )

Proof

Step	Hyp	Ref	Expression
1		sb6x.1	|- F/ x ph
2	1	sbf	|- ( [ y / x ] ph <-> ph )
3		biidd	|- ( x = y -> ( ph <-> ph ) )
4	1 3	equsal	|- ( A. x ( x = y -> ph ) <-> ph )
5	2 4	bitr4i	|- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )