Theorem exdistrf 1971

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.)

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		biidd 228	. . . . 5 ⊢ (∀x x = y → ((φ ∧ ψ) ↔ (φ ∧ ψ)))
2	1	drex1 1967	. . . 4 ⊢ (∀x x = y → (∃x(φ ∧ ψ) ↔ ∃y(φ ∧ ψ)))
3	2	drex2 1968	. . 3 ⊢ (∀x x = y → (∃x∃x(φ ∧ ψ) ↔ ∃x∃y(φ ∧ ψ)))
4		nfe1 1732	. . . . 5 ⊢ Ⅎx∃x(φ ∧ ψ)
5	4	19.9 1783	. . . 4 ⊢ (∃x∃x(φ ∧ ψ) ↔ ∃x(φ ∧ ψ))
6		19.8a 1756	. . . . . 6 ⊢ (ψ → ∃yψ)
7	6	anim2i 552	. . . . 5 ⊢ ((φ ∧ ψ) → (φ ∧ ∃yψ))
8	7	eximi 1576	. . . 4 ⊢ (∃x(φ ∧ ψ) → ∃x(φ ∧ ∃yψ))
9	5, 8	sylbi 187	. . 3 ⊢ (∃x∃x(φ ∧ ψ) → ∃x(φ ∧ ∃yψ))
10	3, 9	syl6bir 220	. 2 ⊢ (∀x x = y → (∃x∃y(φ ∧ ψ) → ∃x(φ ∧ ∃yψ)))
11		nfnae 1956	. . 3 ⊢ Ⅎx ¬ ∀x x = y
12		19.40 1609	. . . 4 ⊢ (∃y(φ ∧ ψ) → (∃yφ ∧ ∃yψ))
13		exdistrf.1	. . . . . 6 ⊢ (¬ ∀x x = y → Ⅎyφ)
14	13	19.9d 1782	. . . . 5 ⊢ (¬ ∀x x = y → (∃yφ → φ))
15	14	anim1d 547	. . . 4 ⊢ (¬ ∀x x = y → ((∃yφ ∧ ∃yψ) → (φ ∧ ∃yψ)))
16	12, 15	syl5 28	. . 3 ⊢ (¬ ∀x x = y → (∃y(φ ∧ ψ) → (φ ∧ ∃yψ)))
17	11, 16	eximd 1770	. 2 ⊢ (¬ ∀x x = y → (∃x∃y(φ ∧ ψ) → ∃x(φ ∧ ∃yψ)))
18	10, 17	pm2.61i 156	1 ⊢ (∃x∃y(φ ∧ ψ) → ∃x(φ ∧ ∃yψ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  oprabid  5551