Theorem reximi 2416

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 9	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	2	reximia 2414	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∈ wcel 1393  ∃wrex 2307

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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-ral 2311  df-rex 2312

This theorem is referenced by:  r19.29d2r  2455  r19.35-1  2460  r19.40  2464  reu3  2731  ssiun  3699  iinss  3708  elunirn  5405  nnawordex  6101  iinerm  6178  erovlem  6198  genprndl  6619  genprndu  6620  appdiv0nq  6662  ltexprlemm  6698  recexsrlem  6859  rereceu  6963  recexre  7569  climi2  9809  climi0  9810  climcaucn  9870  bj-findis  10104