Theorem sbfALT 2593

Description: Alternate version of sbf 2270. (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.ph	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
sbfALT.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
sbfALT	⊢ (𝜃 ↔ 𝜑)

Proof of Theorem sbfALT

Step	Hyp	Ref	Expression
1		sbfALT.1	. 2 ⊢ Ⅎ𝑥𝜑
2		dfsb1.ph	. . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
3	2	sbftALT 2592	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜃 ↔ 𝜑))
4	1, 3	ax-mp 5	1 ⊢ (𝜃 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-nf 1784

This theorem is referenced by:  sbrimALT  2608  sbieALT  2612