Theorem reximia 2720

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 2719	. 2 ⊢ (∀x ∈ A (φ → ψ) → (∃x ∈ A φ → ∃x ∈ A ψ))
2		reximia.1	. 2 ⊢ (x ∈ A → (φ → ψ))
3	1, 2	mprg 2684	1 ⊢ (∃x ∈ A φ → ∃x ∈ A ψ)

Syntax hints:   → wi 4   ∈ wcel 1710  ∃wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-ral 2620  df-rex 2621

This theorem is referenced by:  reximi  2722