Theorem 19.32 765

Description: Theorem 19.32 of [Margaris] p. 90.

Hypothesis

Ref	Expression
19.32.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
19.32	⊢ (∀x(φ ∨ ψ) ↔ (φ ∨ ∀xψ))

Proof of Theorem 19.32

Step	Hyp	Ref	Expression
1		19.32.1	. . . 4 ⊢ (φ → ∀xφ)
2	1	hbne 699	. . 3 ⊢ (¬ φ → ∀x ¬ φ)
3	2	19.21 738	. 2 ⊢ (∀x(¬ φ → ψ) ↔ (¬ φ → ∀xψ))
4		df-or 197	. . 3 ⊢ ((φ ∨ ψ) ↔ (¬ φ → ψ))
5	4	bial 695	. 2 ⊢ (∀x(φ ∨ ψ) ↔ ∀x(¬ φ → ψ))
6		df-or 197	. 2 ⊢ ((φ ∨ ∀xψ) ↔ (¬ φ → ∀xψ))
7	3, 5, 6	3bitr4 158	1 ⊢ (∀x(φ ∨ ψ) ↔ (φ ∨ ∀xψ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195  ∀wal 672

This theorem is referenced by:  19.31 766  2eu3 1069

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

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