Theorem hbor 703

Description: If x is not free in φ and ψ, it is not free in (φ ∨ ψ).

Hypotheses

Ref	Expression
hb.1	⊢ (φ → ∀xφ)
hb.2	⊢ (ψ → ∀xψ)

Assertion

Ref	Expression
hbor	⊢ ((φ ∨ ψ) → ∀x(φ ∨ ψ))

Proof of Theorem hbor

Step	Hyp	Ref	Expression
1		hb.1	. . . 4 ⊢ (φ → ∀xφ)
2	1	hbne 699	. . 3 ⊢ (¬ φ → ∀x ¬ φ)
3		hb.2	. . 3 ⊢ (ψ → ∀xψ)
4	2, 3	hbim 702	. 2 ⊢ ((¬ φ → ψ) → ∀x(¬ φ → ψ))
5		df-or 197	. 2 ⊢ ((φ ∨ ψ) ↔ (¬ φ → ψ))
6	5	bial 695	. 2 ⊢ (∀x(φ ∨ ψ) ↔ ∀x(¬ φ → ψ))
7	4, 5, 6	3imtr4 192	1 ⊢ ((φ ∨ ψ) → ∀x(φ ∨ ψ))

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

This theorem is referenced by:  hb3or 706  hbun 1614  hbpr 1824  hbsuc 2294

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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