Theorem equsal

Description: An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See equsalvw and equsalv for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsex . (Contributed by NM, 2-Jun-1993) (Proof shortened by Andrew Salmon, 12-Aug-2011) (Revised by Mario Carneiro, 3-Oct-2016) (Proof shortened by Wolf Lammen, 5-Feb-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	equsal.1	|- F/ x ps
	equsal.2	|- ( x = y -> ( ph <-> ps ) )
Assertion	equsal	|- ( A. x ( x = y -> ph ) <-> ps )

Proof

Step	Hyp	Ref	Expression
1		equsal.1	|- F/ x ps
2		equsal.2	|- ( x = y -> ( ph <-> ps ) )
3	1	19.23	|- ( A. x ( x = y -> ps ) <-> ( E. x x = y -> ps ) )
4	2	pm5.74i	|- ( ( x = y -> ph ) <-> ( x = y -> ps ) )
5	4	albii	|- ( A. x ( x = y -> ph ) <-> A. x ( x = y -> ps ) )
6		ax6e	|- E. x x = y
7	6	a1bi	|- ( ps <-> ( E. x x = y -> ps ) )
8	3 5 7	3bitr4i	|- ( A. x ( x = y -> ph ) <-> ps )