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Theorem rexlimdvaa 2451
Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)
Hypothesis
Ref Expression
rexlimdvaa.1 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)
Assertion
Ref Expression
rexlimdvaa (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Distinct variable groups:   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem rexlimdvaa
StepHypRef Expression
1 rexlimdvaa.1 . . 3 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)
21expr 361 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
32rexlimdva 2450 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328  df-rex 2329
This theorem is referenced by:  rexlimddv  2454  mulgt0sr  6920  cnegex  7252  receuap  7724  rexanuz  9815  climcaucn  10101  dvdsval2  10111
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