Theorem 19.23t 1118

Description: Closed form of Theorem 19.23 of [Margaris] p. 90.

Assertion

Ref	Expression
19.23t	⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ψ) ↔ (∃xφ → ψ)))

Proof of Theorem 19.23t

Step	Hyp	Ref	Expression
1		hba1 1005	. . 3 ⊢ (∀x(ψ → ∀xψ) → ∀x∀x(ψ → ∀xψ))
2		ax-4 975	. . . . 5 ⊢ (∀xψ → ψ)
3		ax-4 975	. . . . 5 ⊢ (∀x(ψ → ∀xψ) → (ψ → ∀xψ))
4	2, 3	impbid2 520	. . . 4 ⊢ (∀x(ψ → ∀xψ) → (∀xψ ↔ ψ))
5	4	imbi2d 614	. . 3 ⊢ (∀x(ψ → ∀xψ) → ((φ → ∀xψ) ↔ (φ → ψ)))
6	1, 5	albid 1106	. 2 ⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ∀xψ) ↔ ∀x(φ → ψ)))
7	4	imbi2d 614	. . 3 ⊢ (∀x(ψ → ∀xψ) → ((∃xφ → ∀xψ) ↔ (∃xφ → ψ)))
8		hba1 1005	. . . 4 ⊢ (∀xψ → ∀x∀xψ)
9	8	19.23 1065	. . 3 ⊢ (∀x(φ → ∀xψ) ↔ (∃xφ → ∀xψ))
10	7, 9	syl5bb 534	. 2 ⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ∀xψ) ↔ (∃xφ → ψ)))
11	6, 10	bitr3d 532	1 ⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ψ) ↔ (∃xφ → ψ)))

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 956  ∃wex 982

This theorem is referenced by:  vtoclegft 1859  sbciegft 1962

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-4 975  ax-5o 977  ax-6o 980

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983

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