Theorem equsexh 2283

Description: An equivalence related to implicit substitution. See equsexhv 2135 for a version with a dv condition which does not require ax-13 2234. (Contributed by NM, 5-Aug-1993.)

Hypotheses

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

Assertion

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

Proof of Theorem equsexh

Step	Hyp	Ref	Expression
1		equsexh.1	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	nf5i 2011	. 2 ⊢ Ⅎ𝑥𝜓
3		equsexh.2	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	equsex 2281	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701

This theorem is referenced by: (None)