ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexim GIF version

Theorem rexim 2430
Description: Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
rexim (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))

Proof of Theorem rexim
StepHypRef Expression
1 df-ral 2328 . . . 4 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
2 simpl 106 . . . . . . 7 ((𝑥𝐴𝜑) → 𝑥𝐴)
32a1i 9 . . . . . 6 ((𝑥𝐴 → (𝜑𝜓)) → ((𝑥𝐴𝜑) → 𝑥𝐴))
4 pm3.31 253 . . . . . 6 ((𝑥𝐴 → (𝜑𝜓)) → ((𝑥𝐴𝜑) → 𝜓))
53, 4jcad 295 . . . . 5 ((𝑥𝐴 → (𝜑𝜓)) → ((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
65alimi 1360 . . . 4 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) → ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
71, 6sylbi 118 . . 3 (∀𝑥𝐴 (𝜑𝜓) → ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
8 exim 1506 . . 3 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)) → (∃𝑥(𝑥𝐴𝜑) → ∃𝑥(𝑥𝐴𝜓)))
97, 8syl 14 . 2 (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥(𝑥𝐴𝜑) → ∃𝑥(𝑥𝐴𝜓)))
10 df-rex 2329 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
11 df-rex 2329 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
129, 10, 113imtr4g 198 1 (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257  wex 1397  wcel 1409  wral 2323  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-ial 1443
This theorem depends on definitions:  df-bi 114  df-ral 2328  df-rex 2329
This theorem is referenced by:  reximia  2431  reximdai  2434  r19.29  2467  reupick2  3250  ss2iun  3699  chfnrn  5305
  Copyright terms: Public domain W3C validator