Theorem sbi2ALT 2607

Description: Alternate version of sbi2 2310. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dfsb1.p5	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
dfsb1.s2	⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
dfsb1.im	⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓))))

Assertion

Ref	Expression
sbi2ALT	⊢ ((𝜃 → 𝜏) → 𝜂)

Proof of Theorem sbi2ALT

Step	Hyp	Ref	Expression
1		dfsb1.p5	. . . 4 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2		biid 263	. . . 4 ⊢ (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)) ↔ ((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)))
3	1, 2	sbnALT 2595	. . 3 ⊢ (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)) ↔ ¬ 𝜃)
4		dfsb1.im	. . . 4 ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓))))
5		pm2.21 123	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
6	2, 4, 5	sbimiALT 2577	. . 3 ⊢ (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)) → 𝜂)
7	3, 6	sylbir 237	. 2 ⊢ (¬ 𝜃 → 𝜂)
8		dfsb1.s2	. . 3 ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
9		ax-1 6	. . 3 ⊢ (𝜓 → (𝜑 → 𝜓))
10	8, 4, 9	sbimiALT 2577	. 2 ⊢ (𝜏 → 𝜂)
11	7, 10	ja 188	1 ⊢ ((𝜃 → 𝜏) → 𝜂)

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:  sbimALT  2608