Theorem hbsb2ALT 2599

Description: Alternate version of hbsb2 2521. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
dfsb1.p3	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

Assertion

Ref	Expression
hbsb2ALT	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜃 → ∀𝑥𝜃))

Proof of Theorem hbsb2ALT

Step	Hyp	Ref	Expression
1		dfsb1.p3	. . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2	1	sb4ALT 2588	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜃 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	1	sb2ALT 2587	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜃)
4	3	axc4i 2341	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥𝜃)
5	2, 4	syl6 35	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜃 → ∀𝑥𝜃))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785

This theorem is referenced by:  nfsb2ALT  2600