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

Theorem exdistrfor 1681
Description: Distribution of existential quantifiers, with a bound-variable hypothesis saying that 𝑦 is not free in 𝜑, but 𝑥 can be free in 𝜑 (and there is no distinct variable condition on 𝑥 and 𝑦). (Contributed by Jim Kingdon, 25-Feb-2018.)
Hypothesis
Ref Expression
exdistrfor.1 (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥𝑦𝜑)
Assertion
Ref Expression
exdistrfor (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Proof of Theorem exdistrfor
StepHypRef Expression
1 exdistrfor.1 . 2 (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥𝑦𝜑)
2 biidd 161 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜓)))
32drex1 1679 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝜑𝜓) ↔ ∃𝑦(𝜑𝜓)))
43drex2 1620 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑥(𝜑𝜓) ↔ ∃𝑥𝑦(𝜑𝜓)))
5 hbe1 1384 . . . . . 6 (∃𝑥(𝜑𝜓) → ∀𝑥𝑥(𝜑𝜓))
6519.9h 1534 . . . . 5 (∃𝑥𝑥(𝜑𝜓) ↔ ∃𝑥(𝜑𝜓))
7 19.8a 1482 . . . . . . 7 (𝜓 → ∃𝑦𝜓)
87anim2i 324 . . . . . 6 ((𝜑𝜓) → (𝜑 ∧ ∃𝑦𝜓))
98eximi 1491 . . . . 5 (∃𝑥(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
106, 9sylbi 114 . . . 4 (∃𝑥𝑥(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
114, 10syl6bir 153 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)))
12 ax-ial 1427 . . . 4 (∀𝑥𝑦𝜑 → ∀𝑥𝑥𝑦𝜑)
13 19.40 1522 . . . . . 6 (∃𝑦(𝜑𝜓) → (∃𝑦𝜑 ∧ ∃𝑦𝜓))
14 19.9t 1533 . . . . . . . 8 (Ⅎ𝑦𝜑 → (∃𝑦𝜑𝜑))
1514biimpd 132 . . . . . . 7 (Ⅎ𝑦𝜑 → (∃𝑦𝜑𝜑))
1615anim1d 319 . . . . . 6 (Ⅎ𝑦𝜑 → ((∃𝑦𝜑 ∧ ∃𝑦𝜓) → (𝜑 ∧ ∃𝑦𝜓)))
1713, 16syl5 28 . . . . 5 (Ⅎ𝑦𝜑 → (∃𝑦(𝜑𝜓) → (𝜑 ∧ ∃𝑦𝜓)))
1817sps 1430 . . . 4 (∀𝑥𝑦𝜑 → (∃𝑦(𝜑𝜓) → (𝜑 ∧ ∃𝑦𝜓)))
1912, 18eximdh 1502 . . 3 (∀𝑥𝑦𝜑 → (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)))
2011, 19jaoi 636 . 2 ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥𝑦𝜑) → (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)))
211, 20ax-mp 7 1 (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wo 629  wal 1241  wnf 1349  wex 1381
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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-nf 1350
This theorem is referenced by:  oprabidlem  5536
  Copyright terms: Public domain W3C validator