Theorem sbieALT 2612

Description: Alternate version of sbie 2543. (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 263	. . . 4 ⊢ (((𝑥 = 𝑦 → 𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑦)) ↔ ((𝑥 = 𝑦 → 𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑦)))
2	1	equsb1ALT 2600	. . 3 ⊢ ((𝑥 = 𝑦 → 𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑦))
3		biid 263	. . . 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 2576	. . 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 263	. . 3 ⊢ (((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
9		sbieALT.1	. . . 4 ⊢ Ⅎ𝑥𝜓
10	8, 9	sbfALT 2593	. . 3 ⊢ (((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) ↔ 𝜓)
11	7, 8, 3, 10	sblbisALT 2611	. 2 ⊢ (((𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ↔ 𝜓))) ↔ (𝜃 ↔ 𝜓))
12	6, 11	mpbi 232	1 ⊢ (𝜃 ↔ 𝜓)

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-10 2144  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784

This theorem is referenced by:  sbiedALT  2613