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Theorem exdistrf 2465
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). Usage of this theorem is discouraged because it depends on ax-13 2386. Check out exdistr 1951 for a version requiring fewer axioms. (Contributed by Mario Carneiro, 20-Mar-2013.) (Proof shortened by Wolf Lammen, 14-May-2018.) (New usage is discouraged.)
Hypothesis
Ref Expression
exdistrf.1 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)
Assertion
Ref Expression
exdistrf (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Proof of Theorem exdistrf
StepHypRef Expression
1 nfe1 2150 . 2 𝑥𝑥(𝜑 ∧ ∃𝑦𝜓)
2 19.8a 2176 . . . . . 6 (𝜓 → ∃𝑦𝜓)
32anim2i 618 . . . . 5 ((𝜑𝜓) → (𝜑 ∧ ∃𝑦𝜓))
43eximi 1831 . . . 4 (∃𝑦(𝜑𝜓) → ∃𝑦(𝜑 ∧ ∃𝑦𝜓))
5 biidd 264 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ((𝜑 ∧ ∃𝑦𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓)))
65drex1 2459 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ ∃𝑦(𝜑 ∧ ∃𝑦𝜓)))
74, 6syl5ibr 248 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)))
8 19.40 1883 . . . 4 (∃𝑦(𝜑𝜓) → (∃𝑦𝜑 ∧ ∃𝑦𝜓))
9 exdistrf.1 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)
10919.9d 2199 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑦𝜑𝜑))
1110anim1d 612 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → ((∃𝑦𝜑 ∧ ∃𝑦𝜓) → (𝜑 ∧ ∃𝑦𝜓)))
12 19.8a 2176 . . . 4 ((𝜑 ∧ ∃𝑦𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
138, 11, 12syl56 36 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)))
147, 13pm2.61i 184 . 2 (∃𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
151, 14exlimi 2213 1 (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wal 1531  wex 1776  wnf 1780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-12 2173  ax-13 2386
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781
This theorem is referenced by:  oprabid  7182
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