Theorem equsexh 2424

Description: An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2375. See equsexhv 2291 for a version with a disjoint variable condition which does not require ax-13 2375. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
equsexh	⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)

Proof of Theorem equsexh

Step	Hyp	Ref	Expression
1		equsalh.1	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	nf5i 2145	. 2 ⊢ Ⅎ𝑥𝜓
3		equsalh.2	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	equsex 2421	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1537  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-12 2176  ax-13 2375

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783

This theorem is referenced by: (None)