Theorem equsexhv 2135

Description: Version of equsexh 2283 with a dv condition, which does not require ax-13 2234. (Contributed by BJ, 31-May-2019.)

Hypotheses

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

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem equsexhv

Step	Hyp	Ref	Expression
1		equsexhv.nf	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	nf5i 2011	. 2 ⊢ Ⅎ𝑥𝜓
3		equsexhv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	equsexv 2095	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

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

This theorem is referenced by:  cleljustALT  2173