Theorem ceqsex 2894

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 470	. . 3 ⊢ ((x = A ∧ φ) → ψ)
4	1, 3	exlimi 1803	. 2 ⊢ (∃x(x = A ∧ φ) → ψ)
5	2	biimprcd 216	. . . 4 ⊢ (ψ → (x = A → φ))
6	1, 5	alrimi 1765	. . 3 ⊢ (ψ → ∀x(x = A → φ))
7		ceqsex.2	. . . 4 ⊢ A ∈ V
8	7	isseti 2866	. . 3 ⊢ ∃x x = A
9		exintr 1614	. . 3 ⊢ (∀x(x = A → φ) → (∃x x = A → ∃x(x = A ∧ φ)))
10	6, 8, 9	ee10 1376	. 2 ⊢ (ψ → ∃x(x = A ∧ φ))
11	4, 10	impbii 180	1 ⊢ (∃x(x = A ∧ φ) ↔ ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  ceqsexv  2895  ceqsex2  2896