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Theorem reximi2 2573
Description: Inference quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 8-Nov-2004.)
Hypothesis
Ref Expression
reximi2.1 ((𝑥𝐴𝜑) → (𝑥𝐵𝜓))
Assertion
Ref Expression
reximi2 (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜓)

Proof of Theorem reximi2
StepHypRef Expression
1 reximi2.1 . . 3 ((𝑥𝐴𝜑) → (𝑥𝐵𝜓))
21eximi 1600 . 2 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥(𝑥𝐵𝜓))
3 df-rex 2461 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
4 df-rex 2461 . 2 (∃𝑥𝐵 𝜓 ↔ ∃𝑥(𝑥𝐵𝜓))
52, 3, 43imtr4i 201 1 (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wex 1492  wcel 2148  wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-rex 2461
This theorem is referenced by:  btwnz  9370  ioo0  10257
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