Theorem ceqsex 2727

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 1574	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓)
5	2	biimprcd 159	. . . 4 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑))
6	1, 5	alrimi 1503	. . 3 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))
7		ceqsex.2	. . . 4 ⊢ 𝐴 ∈ V
8	7	isseti 2697	. . 3 ⊢ ∃𝑥 𝑥 = 𝐴
9		exintr 1614	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
10	6, 8, 9	mpisyl 1423	. 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 1330   = wceq 1332  Ⅎwnf 1437  ∃wex 1469   ∈ wcel 1481  Vcvv 2689

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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691

This theorem is referenced by:  ceqsexv  2728  ceqsex2  2729