Theorem reximia 2486

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)

Hypothesis

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

Assertion

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

Proof of Theorem reximia

Step	Hyp	Ref	Expression
1		rexim 2485	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
2		reximia.1	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	1, 2	mprg 2448	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∈ wcel 1448  ∃wrex 2376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-4 1455  ax-ial 1482

This theorem depends on definitions:  df-bi 116  df-ral 2380  df-rex 2381

This theorem is referenced by:  reximi  2488  iunpw  4339  nsmallnqq  7121  1idprl  7299  1idpru  7300  qmulz  9265  zq  9268  caubnd2  10729