Theorem exdistrfor 1664

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 161	. . . . . 6 ⊢ (∀x x = y → ((φ ∧ ψ) ↔ (φ ∧ ψ)))
3	2	drex1 1662	. . . . 5 ⊢ (∀x x = y → (∃x(φ ∧ ψ) ↔ ∃y(φ ∧ ψ)))
4	3	drex2 1603	. . . 4 ⊢ (∀x x = y → (∃x∃x(φ ∧ ψ) ↔ ∃x∃y(φ ∧ ψ)))
5		hbe1 1366	. . . . . 6 ⊢ (∃x(φ ∧ ψ) → ∀x∃x(φ ∧ ψ))
6	5	19.9h 1518	. . . . 5 ⊢ (∃x∃x(φ ∧ ψ) ↔ ∃x(φ ∧ ψ))
7		19.8a 1466	. . . . . . 7 ⊢ (ψ → ∃yψ)
8	7	anim2i 324	. . . . . 6 ⊢ ((φ ∧ ψ) → (φ ∧ ∃yψ))
9	8	eximi 1475	. . . . 5 ⊢ (∃x(φ ∧ ψ) → ∃x(φ ∧ ∃yψ))
10	6, 9	sylbi 114	. . . 4 ⊢ (∃x∃x(φ ∧ ψ) → ∃x(φ ∧ ∃yψ))
11	4, 10	syl6bir 153	. . 3 ⊢ (∀x x = y → (∃x∃y(φ ∧ ψ) → ∃x(φ ∧ ∃yψ)))
12		ax-ial 1410	. . . 4 ⊢ (∀xℲyφ → ∀x∀xℲyφ)
13		19.40 1506	. . . . . 6 ⊢ (∃y(φ ∧ ψ) → (∃yφ ∧ ∃yψ))
14		19.9t 1517	. . . . . . . 8 ⊢ (Ⅎyφ → (∃yφ ↔ φ))
15	14	biimpd 132	. . . . . . 7 ⊢ (Ⅎyφ → (∃yφ → φ))
16	15	anim1d 319	. . . . . 6 ⊢ (Ⅎyφ → ((∃yφ ∧ ∃yψ) → (φ ∧ ∃yψ)))
17	13, 16	syl5 28	. . . . 5 ⊢ (Ⅎyφ → (∃y(φ ∧ ψ) → (φ ∧ ∃yψ)))
18	17	sps 1413	. . . 4 ⊢ (∀xℲyφ → (∃y(φ ∧ ψ) → (φ ∧ ∃yψ)))
19	12, 18	eximdh 1486	. . 3 ⊢ (∀xℲyφ → (∃x∃y(φ ∧ ψ) → ∃x(φ ∧ ∃yψ)))
20	11, 19	jaoi 623	. 2 ⊢ ((∀x x = y ∨ ∀xℲyφ) → (∃x∃y(φ ∧ ψ) → ∃x(φ ∧ ∃yψ)))
21	1, 20	ax-mp 7	1 ⊢ (∃x∃y(φ ∧ ψ) → ∃x(φ ∧ ∃yψ))

Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 616  ∀wal 1226  Ⅎwnf 1329  ∃wex 1363

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410

This theorem depends on definitions:  df-bi 110  df-nf 1330

This theorem is referenced by:  oprabidlem  5458