Theorem 19.21g 792

Description: Closed form of Theorem 19.21 of [Margaris] p. 90.

Assertion

Ref	Expression
19.21g	⊢ (∀x(φ → ∀xφ) → (∀x(φ → ψ) ↔ (φ → ∀xψ)))

Proof of Theorem 19.21g

Step	Hyp	Ref	Expression
1		19.20 690	. . . . 5 ⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ))
2	1	syl3d 26	. . . 4 ⊢ (∀x(φ → ψ) → ((φ → ∀xφ) → (φ → ∀xψ)))
3	2	com12 13	. . 3 ⊢ ((φ → ∀xφ) → (∀x(φ → ψ) → (φ → ∀xψ)))
4	3	a4s 682	. 2 ⊢ (∀x(φ → ∀xφ) → (∀x(φ → ψ) → (φ → ∀xψ)))
5		hba1 698	. . . 4 ⊢ (∀x(φ → ∀xφ) → ∀x∀x(φ → ∀xφ))
6		ax-4 673	. . . 4 ⊢ (∀x(φ → ∀xφ) → (φ → ∀xφ))
7		hba1 698	. . . . 5 ⊢ (∀xψ → ∀x∀xψ)
8	7	a1i 7	. . . 4 ⊢ (∀x(φ → ∀xφ) → (∀xψ → ∀x∀xψ))
9	5, 6, 8	hbimd 787	. . 3 ⊢ (∀x(φ → ∀xφ) → ((φ → ∀xψ) → ∀x(φ → ∀xψ)))
10		ax-4 673	. . . . 5 ⊢ (∀xψ → ψ)
11	10	syl3 18	. . . 4 ⊢ ((φ → ∀xψ) → (φ → ψ))
12	11	19.20i 691	. . 3 ⊢ (∀x(φ → ∀xψ) → ∀x(φ → ψ))
13	9, 12	syl6 23	. 2 ⊢ (∀x(φ → ∀xφ) → ((φ → ∀xψ) → ∀x(φ → ψ)))
14	4, 13	impbid 397	1 ⊢ (∀x(φ → ∀xφ) → (∀x(φ → ψ) ↔ (φ → ∀xψ)))

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

This theorem is referenced by:  sbcom 916  sbal2 1005

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-an 198

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