Theorem 19.31 766

Description: Theorem 19.31 of [Margaris] p. 90.

Hypothesis

Ref	Expression
19.31.1	⊢ (ψ → ∀xψ)

Assertion

Ref	Expression
19.31	⊢ (∀x(φ ∨ ψ) ↔ (∀xφ ∨ ψ))

Proof of Theorem 19.31

Step	Hyp	Ref	Expression
1		19.31.1	. . 3 ⊢ (ψ → ∀xψ)
2	1	19.32 765	. 2 ⊢ (∀x(ψ ∨ φ) ↔ (ψ ∨ ∀xφ))
3		orcom 209	. . 3 ⊢ ((φ ∨ ψ) ↔ (ψ ∨ φ))
4	3	bial 695	. 2 ⊢ (∀x(φ ∨ ψ) ↔ ∀x(ψ ∨ φ))
5		orcom 209	. 2 ⊢ ((∀xφ ∨ ψ) ↔ (ψ ∨ ∀xφ))
6	2, 4, 5	3bitr4 158	1 ⊢ (∀x(φ ∨ ψ) ↔ (∀xφ ∨ ψ))

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

This theorem is referenced by:  19.41 774  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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