Theorem eeor 795

Description: Rearrange existential quantifiers.

Hypotheses

Ref	Expression
eeor.1	⊢ (φ → ∀yφ)
eeor.2	⊢ (ψ → ∀xψ)

Assertion

Ref	Expression
eeor	⊢ (∃x∃y(φ ∨ ψ) ↔ (∃xφ ∨ ∃yψ))

Proof of Theorem eeor

Step	Hyp	Ref	Expression
1		eeor.1	. . . 4 ⊢ (φ → ∀yφ)
2	1	19.45 769	. . 3 ⊢ (∃y(φ ∨ ψ) ↔ (φ ∨ ∃yψ))
3	2	biex 733	. 2 ⊢ (∃x∃y(φ ∨ ψ) ↔ ∃x(φ ∨ ∃yψ))
4		eeor.2	. . . 4 ⊢ (ψ → ∀xψ)
5	4	hbex 701	. . 3 ⊢ (∃yψ → ∀x∃yψ)
6	5	19.44 768	. 2 ⊢ (∃x(φ ∨ ∃yψ) ↔ (∃xφ ∨ ∃yψ))
7	3, 6	bitr 151	1 ⊢ (∃x∃y(φ ∨ ψ) ↔ (∃xφ ∨ ∃yψ))

Syntax hints:   → wi 2   ↔ wb 127   ∨ wo 195  ∀wal 672  ∃wex 678

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679

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