Theorem equsb1ALT 2577

Description: Alternate version of equsb1 2509. (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 2563	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑦) → 𝜃)
3		id 22	. 2 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
4	2, 3	mpg 1799	1 ⊢ 𝜃

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∃wex 1781

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

This theorem is referenced by:  sbieALT  2589