Theorem hbexOLD 2138

Description: Obsolete proof of hbex 2142 as of 16-Oct-2021. (Contributed by NM, 12-Mar-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
hbexOLD.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
hbexOLD	⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)

Proof of Theorem hbexOLD

Step	Hyp	Ref	Expression
1		df-ex 1696	. 2 ⊢ (∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑)
2		hbexOLD.1	. . . . 5 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbn 2131	. . . 4 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
4	3	hbal 2023	. . 3 ⊢ (∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑)
5	4	hbn 2131	. 2 ⊢ (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)
6	1, 5	hbxfrbi 1742	1 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701

This theorem is referenced by:  nfexOLD  2141