Theorem exdistrf 2458

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 2379. Check out exdistr 1955 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	⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Proof of Theorem exdistrf

Step	Hyp	Ref	Expression
1		nfe1 2151	. 2 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ ∃𝑦𝜓)
2		19.8a 2178	. . . . . 6 ⊢ (𝜓 → ∃𝑦𝜓)
3	2	anim2i 619	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ ∃𝑦𝜓))
4	3	eximi 1836	. . . 4 ⊢ (∃𝑦(𝜑 ∧ 𝜓) → ∃𝑦(𝜑 ∧ ∃𝑦𝜓))
5		biidd 265	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 ∧ ∃𝑦𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓)))
6	5	drex1 2452	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ ∃𝑦(𝜑 ∧ ∃𝑦𝜓)))
7	4, 6	syl5ibr 249	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)))
8		19.40 1887	. . . 4 ⊢ (∃𝑦(𝜑 ∧ 𝜓) → (∃𝑦𝜑 ∧ ∃𝑦𝜓))
9		exdistrf.1	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)
10	9	19.9d 2201	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑦𝜑 → 𝜑))
11	10	anim1d 613	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((∃𝑦𝜑 ∧ ∃𝑦𝜓) → (𝜑 ∧ ∃𝑦𝜓)))
12		19.8a 2178	. . . 4 ⊢ ((𝜑 ∧ ∃𝑦𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
13	8, 11, 12	syl56 36	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)))
14	7, 13	pm2.61i 185	. 2 ⊢ (∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
15	1, 14	exlimi 2215	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786

This theorem is referenced by:  oprabid  7167