Theorem 19.21h 1551

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". New proofs should use 19.21 1577 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Hypothesis

Ref	Expression
19.21h.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

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

Proof of Theorem 19.21h

Step	Hyp	Ref	Expression
1		19.21h.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		alim 1451	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
3	1, 2	syl5 32	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
4		hba1 1534	. . . 4 ⊢ (∀𝑥𝜓 → ∀𝑥∀𝑥𝜓)
5	1, 4	hbim 1539	. . 3 ⊢ ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓))
6		ax-4 1504	. . . 4 ⊢ (∀𝑥𝜓 → 𝜓)
7	6	imim2i 12	. . 3 ⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓))
8	5, 7	alrimih 1463	. 2 ⊢ ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
9	3, 8	impbii 125	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 106  ax-ia3 107  ax-5 1441  ax-gen 1443  ax-4 1504  ax-ial 1528  ax-i5r 1529

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  hbim1  1564  nf3  1663  19.21v  1867