Theorem equsalh 2425

Description: An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2377. See equsalhw 2292 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 2-Jun-1993.) (New usage is discouraged.)

Hypotheses

Ref	Expression
equsalh.1	⊢ (𝜓 → ∀𝑥𝜓)
equsalh.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
equsalh	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

Proof of Theorem equsalh

Step	Hyp	Ref	Expression
1		equsalh.1	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	nf5i 2147	. 2 ⊢ Ⅎ𝑥𝜓
3		equsalh.2	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	equsal 2422	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-12 2178  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784

This theorem is referenced by:  dvelimf-o  38952