Metamath Proof Explorer

Theorem sb3b

Description: Simplified definition of substitution when variables are distinct. This is the biconditional strengthening of sb3 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by BJ, 6-Oct-2018) Shorten sb3 . (Revised by Wolf Lammen, 21-Feb-2021) (New usage is discouraged.)

Ref Expression
Assertion sb3b ¬ x x = y y x φ x x = y φ


Step Hyp Ref Expression
1 sb4b ¬ x x = y y x φ x x = y φ
2 equs5 ¬ x x = y x x = y φ x x = y φ
3 1 2 bitr4d ¬ x x = y y x φ x x = y φ