Theorem ceqsex 1370

Description: Elimination of an existential quantifier, using implicit substitution.

Hypotheses

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

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem ceqsex

Step	Hyp	Ref	Expression
1		ceqsex.1	. . 3 ⊢ (ψ → ∀xψ)
2		ceqsex.3	. . . 4 ⊢ (x = A → (φ ↔ ψ))
3	2	biimpa 324	. . 3 ⊢ ((x = A ∧ φ) → ψ)
4	1, 3	19.23ai 746	. 2 ⊢ (∃x(x = A ∧ φ) → ψ)
5		ceqsex.2	. . . 4 ⊢ A ∈ V
6	5	isseti 1352	. . 3 ⊢ ∃x x = A
7	2	biimprcd 138	. . . . 5 ⊢ (ψ → (x = A → φ))
8	1, 7	19.21ai 740	. . . 4 ⊢ (ψ → ∀x(x = A → φ))
9		exintr 793	. . . 4 ⊢ (∀x(x = A → φ) → (∃x x = A → ∃x(x = A ∧ φ)))
10	8, 9	syl 12	. . 3 ⊢ (ψ → (∃x x = A → ∃x(x = A ∧ φ)))
11	6, 10	mpi 44	. 2 ⊢ (ψ → ∃x(x = A ∧ φ))
12	4, 11	impbi 139	1 ⊢ (∃x(x = A ∧ φ) ↔ ψ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  ceqsexv 1371

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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