Theorem sbrimALT 2609

Description: Alternate version of sbrim 2313. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dfsb1.s3	⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
dfsb1.i2	⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓))))
sbrimALT.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
sbrimALT	⊢ (𝜂 ↔ (𝜑 → 𝜏))

Proof of Theorem sbrimALT

Step	Hyp	Ref	Expression
1		biid 263	. . 3 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2		dfsb1.s3	. . 3 ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
3		dfsb1.i2	. . 3 ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓))))
4	1, 2, 3	sbimALT 2608	. 2 ⊢ (𝜂 ↔ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → 𝜏))
5		sbrimALT.1	. . . 4 ⊢ Ⅎ𝑥𝜑
6	1, 5	sbfALT 2594	. . 3 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ 𝜑)
7	6	imbi1i 352	. 2 ⊢ ((((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → 𝜏) ↔ (𝜑 → 𝜏))
8	4, 7	bitri 277	1 ⊢ (𝜂 ↔ (𝜑 → 𝜏))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∃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-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:  sbiedALT  2614