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	⊢ Ⅎ 𝑥 𝜓
	equsal.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	equsal	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		equsal.1	⊢ Ⅎ 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 )