Theorem exdistrfor 1772

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 ∨ ∀𝑥Ⅎ𝑦𝜑)

Assertion

Ref	Expression
exdistrfor	⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Proof of Theorem exdistrfor

Step	Hyp	Ref	Expression
1		exdistrfor.1	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥Ⅎ𝑦𝜑)
2		biidd 171	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜓)))
3	2	drex1 1770	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑦(𝜑 ∧ 𝜓)))
4	3	drex2 1710	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦(𝜑 ∧ 𝜓)))
5		hbe1 1471	. . . . . 6 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∀𝑥∃𝑥(𝜑 ∧ 𝜓))
6	5	19.9h 1622	. . . . 5 ⊢ (∃𝑥∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓))
7		19.8a 1569	. . . . . . 7 ⊢ (𝜓 → ∃𝑦𝜓)
8	7	anim2i 339	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ ∃𝑦𝜓))
9	8	eximi 1579	. . . . 5 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
10	6, 9	sylbi 120	. . . 4 ⊢ (∃𝑥∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
11	4, 10	syl6bir 163	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)))
12		ax-ial 1514	. . . 4 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥∀𝑥Ⅎ𝑦𝜑)
13		19.40 1610	. . . . . 6 ⊢ (∃𝑦(𝜑 ∧ 𝜓) → (∃𝑦𝜑 ∧ ∃𝑦𝜓))
14		19.9t 1621	. . . . . . . 8 ⊢ (Ⅎ𝑦𝜑 → (∃𝑦𝜑 ↔ 𝜑))
15	14	biimpd 143	. . . . . . 7 ⊢ (Ⅎ𝑦𝜑 → (∃𝑦𝜑 → 𝜑))
16	15	anim1d 334	. . . . . 6 ⊢ (Ⅎ𝑦𝜑 → ((∃𝑦𝜑 ∧ ∃𝑦𝜓) → (𝜑 ∧ ∃𝑦𝜓)))
17	13, 16	syl5 32	. . . . 5 ⊢ (Ⅎ𝑦𝜑 → (∃𝑦(𝜑 ∧ 𝜓) → (𝜑 ∧ ∃𝑦𝜓)))
18	17	sps 1517	. . . 4 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑦(𝜑 ∧ 𝜓) → (𝜑 ∧ ∃𝑦𝜓)))
19	12, 18	eximdh 1590	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)))
20	11, 19	jaoi 705	. 2 ⊢ ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥Ⅎ𝑦𝜑) → (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)))
21	1, 20	ax-mp 5	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 697  ∀wal 1329  Ⅎwnf 1436  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1437

This theorem is referenced by:  oprabidlem  5802