Theorem nfsb4ALT 2605

Description: Alternate version of nfsb4 2540. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dfsb1.p5	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
nfsb4ALT.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
nfsb4ALT	⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃)

Proof of Theorem nfsb4ALT

Step	Hyp	Ref	Expression
1		dfsb1.p5	. . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2	1	nfsb4tALT 2604	. 2 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃))
3		nfsb4ALT.1	. 2 ⊢ Ⅎ𝑧𝜑
4	2, 3	mpg 1798	1 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780  Ⅎwnf 1784

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-11 2161  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785

This theorem is referenced by:  sbco2ALT  2615