Theorem hbim1OLD 2227

Description: Obsolete proof of hbim 2127 as of 6-Oct-2021. (Contributed by NM, 2-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hbim1OLD.1	⊢ (𝜑 → ∀𝑥𝜑)
hbim1OLD.2	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Assertion

Ref	Expression
hbim1OLD	⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))

Proof of Theorem hbim1OLD

Step	Hyp	Ref	Expression
1		hbim1OLD.2	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
2	1	a2i 14	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
3		hbim1OLD.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
4	3	19.21hOLD 2216	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
5	2, 4	sylibr 224	1 ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nfOLD 1721

This theorem is referenced by:  nfim1OLD  2228  hbimOLD  2231