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Theorem exdistrfor 1723
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 170 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜓)))
32drex1 1721 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝜑𝜓) ↔ ∃𝑦(𝜑𝜓)))
43drex2 1662 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑥(𝜑𝜓) ↔ ∃𝑥𝑦(𝜑𝜓)))
5 hbe1 1425 . . . . . 6 (∃𝑥(𝜑𝜓) → ∀𝑥𝑥(𝜑𝜓))
6519.9h 1575 . . . . 5 (∃𝑥𝑥(𝜑𝜓) ↔ ∃𝑥(𝜑𝜓))
7 19.8a 1523 . . . . . . 7 (𝜓 → ∃𝑦𝜓)
87anim2i 334 . . . . . 6 ((𝜑𝜓) → (𝜑 ∧ ∃𝑦𝜓))
98eximi 1532 . . . . 5 (∃𝑥(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
106, 9sylbi 119 . . . 4 (∃𝑥𝑥(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
114, 10syl6bir 162 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)))
12 ax-ial 1468 . . . 4 (∀𝑥𝑦𝜑 → ∀𝑥𝑥𝑦𝜑)
13 19.40 1563 . . . . . 6 (∃𝑦(𝜑𝜓) → (∃𝑦𝜑 ∧ ∃𝑦𝜓))
14 19.9t 1574 . . . . . . . 8 (Ⅎ𝑦𝜑 → (∃𝑦𝜑𝜑))
1514biimpd 142 . . . . . . 7 (Ⅎ𝑦𝜑 → (∃𝑦𝜑𝜑))
1615anim1d 329 . . . . . 6 (Ⅎ𝑦𝜑 → ((∃𝑦𝜑 ∧ ∃𝑦𝜓) → (𝜑 ∧ ∃𝑦𝜓)))
1713, 16syl5 32 . . . . 5 (Ⅎ𝑦𝜑 → (∃𝑦(𝜑𝜓) → (𝜑 ∧ ∃𝑦𝜓)))
1817sps 1471 . . . 4 (∀𝑥𝑦𝜑 → (∃𝑦(𝜑𝜓) → (𝜑 ∧ ∃𝑦𝜓)))
1912, 18eximdh 1543 . . 3 (∀𝑥𝑦𝜑 → (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)))
2011, 19jaoi 669 . 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 102  wo 662  wal 1283  wnf 1390  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  oprabidlem  5613
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