Theorem reximdva0m 3230

Description: Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)

Hypothesis

Ref	Expression
reximdva0m.1	⊢ ((φ ∧ x ∈ A) → ψ)

Assertion

Ref	Expression
reximdva0m	⊢ ((φ ∧ ∃x x ∈ A) → ∃x ∈ A ψ)

Distinct variable groups:   x,A   φ,x

Allowed substitution hint:   ψ(x)

Proof of Theorem reximdva0m

Step	Hyp	Ref	Expression
1		reximdva0m.1	. . . . . 6 ⊢ ((φ ∧ x ∈ A) → ψ)
2	1	ex 108	. . . . 5 ⊢ (φ → (x ∈ A → ψ))
3	2	ancld 308	. . . 4 ⊢ (φ → (x ∈ A → (x ∈ A ∧ ψ)))
4	3	eximdv 1757	. . 3 ⊢ (φ → (∃x x ∈ A → ∃x(x ∈ A ∧ ψ)))
5	4	imp 115	. 2 ⊢ ((φ ∧ ∃x x ∈ A) → ∃x(x ∈ A ∧ ψ))
6		df-rex 2306	. 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ))
7	5, 6	sylibr 137	1 ⊢ ((φ ∧ ∃x x ∈ A) → ∃x ∈ A ψ)

Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-rex 2306

This theorem is referenced by: (None)