Theorem sbieALT 2589

Description: Alternate version of sbie 2521. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 13-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dfsb1.p7	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
sbieALT.1	⊢ Ⅎ𝑥𝜓
sbieALT.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbieALT	⊢ (𝜃 ↔ 𝜓)

Proof of Theorem sbieALT

Step	Hyp	Ref	Expression
1		biid 264	. . . 4 ⊢ (((𝑥 = 𝑦 → 𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑦)) ↔ ((𝑥 = 𝑦 → 𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑦)))
2	1	equsb1ALT 2577	. . 3 ⊢ ((𝑥 = 𝑦 → 𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑦))
3		biid 264	. . . 4 ⊢ (((𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ↔ 𝜓))) ↔ ((𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ↔ 𝜓))))
4		sbieALT.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 3, 4	sbimiALT 2553	. . 3 ⊢ (((𝑥 = 𝑦 → 𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑦)) → ((𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ↔ 𝜓))))
6	2, 5	ax-mp 5	. 2 ⊢ ((𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ↔ 𝜓)))
7		dfsb1.p7	. . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
8		biid 264	. . 3 ⊢ (((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
9		sbieALT.1	. . . 4 ⊢ Ⅎ𝑥𝜓
10	8, 9	sbfALT 2570	. . 3 ⊢ (((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) ↔ 𝜓)
11	7, 8, 3, 10	sblbisALT 2588	. 2 ⊢ (((𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ↔ 𝜓))) ↔ (𝜃 ↔ 𝜓))
12	6, 11	mpbi 233	1 ⊢ (𝜃 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∃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-10 2142  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786

This theorem is referenced by:  sbiedALT  2590