Theorem 19.45 769

Description: Theorem 19.45 of [Margaris] p. 90.

Hypothesis

Ref	Expression
19.45.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
19.45	⊢ (∃x(φ ∨ ψ) ↔ (φ ∨ ∃xψ))

Proof of Theorem 19.45

Step	Hyp	Ref	Expression
1		19.43 767	. 2 ⊢ (∃x(φ ∨ ψ) ↔ (∃xφ ∨ ∃xψ))
2		19.45.1	. . . 4 ⊢ (φ → ∀xφ)
3	2	19.9r 718	. . 3 ⊢ (φ ↔ ∃xφ)
4	3	orbi1i 215	. 2 ⊢ ((φ ∨ ∃xψ) ↔ (∃xφ ∨ ∃xψ))
5	1, 4	bitr4 154	1 ⊢ (∃x(φ ∨ ψ) ↔ (φ ∨ ∃xψ))

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

This theorem is referenced by:  eeor 795  iununi 2037

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-gen 677

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

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