Metamath Proof Explorer

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 x ψ
equsal.2 x = y φ ψ
Assertion equsal x x = y φ ψ


Step Hyp Ref Expression
1 equsal.1 x ψ
2 equsal.2 x = y φ ψ
3 1 19.23 x x = y ψ x x = y ψ
4 2 pm5.74i x = y φ x = y ψ
5 4 albii x x = y φ x x = y ψ
6 ax6e x x = y
7 6 a1bi ψ x x = y ψ
8 3 5 7 3bitr4i x x = y φ ψ