Theorem equsb1ALT 2601

Description: Alternate version of equsb1 2530. (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
dfsb1.p4	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑦)))

Assertion

Ref	Expression
equsb1ALT	⊢ 𝜃

Proof of Theorem equsb1ALT

Step	Hyp	Ref	Expression
1		dfsb1.p4	. . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑦)))
2	1	sb2ALT 2587	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑦) → 𝜃)
3		id 22	. 2 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
4	2, 3	mpg 1798	1 ⊢ 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  ax-13 2390

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

This theorem is referenced by:  sbieALT  2613