Theorem equsex 1742

Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Hypotheses

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

Assertion

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

Proof of Theorem equsex

Step	Hyp	Ref	Expression
1		equsex.1	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
2		equsex.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	biimpa 296	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜓)
4	1, 3	exlimih 1607	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜓)
5		a9e 1710	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
6		idd 21	. . . . 5 ⊢ (𝜓 → (𝑥 = 𝑦 → 𝑥 = 𝑦))
7	2	biimprcd 160	. . . . 5 ⊢ (𝜓 → (𝑥 = 𝑦 → 𝜑))
8	6, 7	jcad 307	. . . 4 ⊢ (𝜓 → (𝑥 = 𝑦 → (𝑥 = 𝑦 ∧ 𝜑)))
9	1, 8	eximdh 1625	. . 3 ⊢ (𝜓 → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
10	5, 9	mpi 15	. 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
11	4, 10	impbii 126	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362   = wceq 1364  ∃wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  cbvexv1  1766  cbvexh  1769  sb56  1900  sb10f  2014  cleljust  2173