Theorem sbco2ALT 2615

Description: Alternate version of sbco2 2553. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dfsb1.p8	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
dfsb1.sb	⊢ (𝜏 ↔ ((𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) ∧ ∃𝑧(𝑧 = 𝑦 ∧ ((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)))))
sbco2ALT.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
sbco2ALT	⊢ (𝜏 ↔ 𝜃)

Proof of Theorem sbco2ALT

Step	Hyp	Ref	Expression
1		dfsb1.sb	. . . . 5 ⊢ (𝜏 ↔ ((𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) ∧ ∃𝑧(𝑧 = 𝑦 ∧ ((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)))))
2	1	sbequ12ALT 2581	. . . 4 ⊢ (𝑧 = 𝑦 → (((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)) ↔ 𝜏))
3		biid 263	. . . . 5 ⊢ (((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)) ↔ ((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)))
4		dfsb1.p8	. . . . 5 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
5	3, 4	sbequALT 2597	. . . 4 ⊢ (𝑧 = 𝑦 → (((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)) ↔ 𝜃))
6	2, 5	bitr3d 283	. . 3 ⊢ (𝑧 = 𝑦 → (𝜏 ↔ 𝜃))
7	6	sps 2184	. 2 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝜏 ↔ 𝜃))
8		nfnae 2456	. . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
9		sbco2ALT.1	. . . 4 ⊢ Ⅎ𝑧𝜑
10	4, 9	nfsb4ALT 2605	. . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃)
11	5	a1i 11	. . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → (𝑧 = 𝑦 → (((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)) ↔ 𝜃)))
12	1, 8, 10, 11	sbiedALT 2614	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → (𝜏 ↔ 𝜃))
13	7, 12	pm2.61i 184	1 ⊢ (𝜏 ↔ 𝜃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780  Ⅎwnf 1784

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-10 2145  ax-11 2161  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785

This theorem is referenced by:  sb7fALT  2616