Theorem sbequ12ALT 2581

Description: Alternate version of sbequ12 2253. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
sbequ12ALT	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜃))

Proof of Theorem sbequ12ALT

Step	Hyp	Ref	Expression
1		dfsb1.ph	. . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2	1	sbequ1ALT 2579	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜃))
3	1	sbequ2ALT 2580	. 2 ⊢ (𝑥 = 𝑦 → (𝜃 → 𝜑))
4	2, 3	impbid 214	1 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜃))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∃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-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by:  nfsb4tALT  2604  sbco2ALT  2615