Theorem hbim1 1570

Description: A closed form of hbim 1545. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
hbim1.1	⊢ (𝜑 → ∀𝑥𝜑)
hbim1.2	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Assertion

Ref	Expression
hbim1	⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))

Proof of Theorem hbim1

Step	Hyp	Ref	Expression
1		hbim1.2	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
2	1	a2i 11	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
3		hbim1.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
4	3	19.21h 1557	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
5	2, 4	sylibr 134	1 ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-4 1510  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  nfim1  1571  sbco2d  1966  sbco2vd  1967