Theorem sbftALT 2569

Description: Alternate version of sbft 2267. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
sbftALT	⊢ (Ⅎ𝑥𝜑 → (𝜃 ↔ 𝜑))

Proof of Theorem sbftALT

Step	Hyp	Ref	Expression
1		dfsb1.ph	. . . 4 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2	1	spsbeALT 2565	. . 3 ⊢ (𝜃 → ∃𝑥𝜑)
3		19.9t 2202	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
4	2, 3	syl5ib 247	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜃 → 𝜑))
5		nf5r 2191	. . 3 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
6	1	stdpc4ALT 2566	. . 3 ⊢ (∀𝑥𝜑 → 𝜃)
7	5, 6	syl6 35	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → 𝜃))
8	4, 7	impbid 215	1 ⊢ (Ⅎ𝑥𝜑 → (𝜃 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786

This theorem is referenced by:  sbfALT  2570