Theorem sbanALT 2610

Description: Alternate version of sban 2086. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dfsb1.p6	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
dfsb1.s4	⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
dfsb1.an	⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 ∧ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ∧ 𝜓))))

Assertion

Ref	Expression
sbanALT	⊢ (𝜂 ↔ (𝜃 ∧ 𝜏))

Proof of Theorem sbanALT

Step	Hyp	Ref	Expression
1		biid 263	. . . 4 ⊢ (((𝑥 = 𝑦 → (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → ¬ 𝜓))) ↔ ((𝑥 = 𝑦 → (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → ¬ 𝜓))))
2		biid 263	. . . 4 ⊢ (((𝑥 = 𝑦 → ¬ (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ (𝜑 → ¬ 𝜓))) ↔ ((𝑥 = 𝑦 → ¬ (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ (𝜑 → ¬ 𝜓))))
3	1, 2	sbnALT 2595	. . 3 ⊢ (((𝑥 = 𝑦 → ¬ (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ (𝜑 → ¬ 𝜓))) ↔ ¬ ((𝑥 = 𝑦 → (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → ¬ 𝜓))))
4		dfsb1.p6	. . . . 5 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
5		biid 263	. . . . 5 ⊢ (((𝑥 = 𝑦 → ¬ 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜓)) ↔ ((𝑥 = 𝑦 → ¬ 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜓)))
6	4, 5, 1	sbimALT 2608	. . . 4 ⊢ (((𝑥 = 𝑦 → (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → ¬ 𝜓))) ↔ (𝜃 → ((𝑥 = 𝑦 → ¬ 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜓))))
7		dfsb1.s4	. . . . . 6 ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
8	7, 5	sbnALT 2595	. . . . 5 ⊢ (((𝑥 = 𝑦 → ¬ 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜓)) ↔ ¬ 𝜏)
9	8	imbi2i 338	. . . 4 ⊢ ((𝜃 → ((𝑥 = 𝑦 → ¬ 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜓))) ↔ (𝜃 → ¬ 𝜏))
10	6, 9	bitri 277	. . 3 ⊢ (((𝑥 = 𝑦 → (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → ¬ 𝜓))) ↔ (𝜃 → ¬ 𝜏))
11	3, 10	xchbinx 336	. 2 ⊢ (((𝑥 = 𝑦 → ¬ (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ (𝜑 → ¬ 𝜓))) ↔ ¬ (𝜃 → ¬ 𝜏))
12		dfsb1.an	. . 3 ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 ∧ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ∧ 𝜓))))
13		df-an 399	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
14	12, 2, 13	sbbiiALT 2578	. 2 ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → ¬ (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ (𝜑 → ¬ 𝜓))))
15		df-an 399	. 2 ⊢ ((𝜃 ∧ 𝜏) ↔ ¬ (𝜃 → ¬ 𝜏))
16	11, 14, 15	3bitr4i 305	1 ⊢ (𝜂 ↔ (𝜃 ∧ 𝜏))

Syntax hints:  ¬ wn 3   → 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-10 2145  ax-12 2177  ax-13 2390

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

This theorem is referenced by:  sbbiALT  2611