Theorem exdistrf 2455

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 Mario Carneiro, 20-Mar-2013.) (Proof shortened by Wolf Lammen, 14-May-2018.) Usage of this theorem is discouraged because it depends on ax-13 2380. Use exdistr 1961 instead. (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 2161	. 2 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ ∃𝑦𝜓)
2		19.8a 2193	. . . . . 6 ⊢ (𝜓 → ∃𝑦𝜓)
3	2	anim2i 623	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ ∃𝑦𝜓))
4	3	eximi 1842	. . . 4 ⊢ (∃𝑦(𝜑 ∧ 𝜓) → ∃𝑦(𝜑 ∧ ∃𝑦𝜓))
5		biidd 263	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 ∧ ∃𝑦𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓)))
6	5	drex1 2449	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ ∃𝑦(𝜑 ∧ ∃𝑦𝜓)))
7	4, 6	imbitrrid 247	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)))
8		19.40 1893	. . . 4 ⊢ (∃𝑦(𝜑 ∧ 𝜓) → (∃𝑦𝜑 ∧ ∃𝑦𝜓))
9		exdistrf.1	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)
10	9	19.9d 2215	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑦𝜑 → 𝜑))
11	10	anim1d 617	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((∃𝑦𝜑 ∧ ∃𝑦𝜓) → (𝜑 ∧ ∃𝑦𝜓)))
12		19.8a 2193	. . . 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 183	. 2 ⊢ (∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
15	1, 14	exlimi 2229	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1545  ∃wex 1786  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-12 2189  ax-13 2380

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791

This theorem is referenced by:  oprabid  7395