Theorem reximi 2721

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)

Hypothesis

Ref	Expression
reximi.1	⊢ (φ → ψ)

Assertion

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

Proof of Theorem reximi

Step	Hyp	Ref	Expression
1		reximi.1	. . 3 ⊢ (φ → ψ)
2	1	a1i 10	. 2 ⊢ (x ∈ A → (φ → ψ))
3	2	reximia 2719	1 ⊢ (∃x ∈ A φ → ∃x ∈ A ψ)

Syntax hints:   → wi 4   ∈ wcel 1710  ∃wrex 2615

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 2619  df-rex 2620

This theorem is referenced by:  r19.40  2762  reu3  3026  2reu5  3044  ssiun  4008  iinss  4017  lefinlteq  4463  sucevenodd  4510  sfinltfin  4535  vfinspsslem1  4550  pw1fin  6169  addlec  6208  nncdiv3  6277