Theorem ceqsex 2764

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)

Hypotheses

Ref	Expression
ceqsex.1	⊢ Ⅎ𝑥𝜓
ceqsex.2	⊢ 𝐴 ∈ V
ceqsex.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ceqsex	⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem ceqsex

Step	Hyp	Ref	Expression
1		ceqsex.1	. . 3 ⊢ Ⅎ𝑥𝜓
2		ceqsex.3	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	biimpa 294	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → 𝜓)
4	1, 3	exlimi 1582	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓)
5	2	biimprcd 159	. . . 4 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑))
6	1, 5	alrimi 1510	. . 3 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))
7		ceqsex.2	. . . 4 ⊢ 𝐴 ∈ V
8	7	isseti 2734	. . 3 ⊢ ∃𝑥 𝑥 = 𝐴
9		exintr 1622	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
10	6, 8, 9	mpisyl 1434	. 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
11	4, 10	impbii 125	1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341   = wceq 1343  Ⅎwnf 1448  ∃wex 1480   ∈ wcel 2136  Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by:  ceqsexv  2765  ceqsex2  2766