Theorem sbbiALT 2587

Description: Alternate version of sbbi 2313. (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.bi	⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ↔ 𝜓))))

Assertion

Ref	Expression
sbbiALT	⊢ (𝜂 ↔ (𝜃 ↔ 𝜏))

Proof of Theorem sbbiALT

Step	Hyp	Ref	Expression
1		dfsb1.bi	. . 3 ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ↔ 𝜓))))
2		biid 264	. . 3 ⊢ (((𝑥 = 𝑦 → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))) ↔ ((𝑥 = 𝑦 → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))))
3		dfbi2 478	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
4	1, 2, 3	sbbiiALT 2554	. 2 ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))))
5		dfsb1.p6	. . . . 5 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
6		dfsb1.s4	. . . . 5 ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
7		biid 264	. . . . 5 ⊢ (((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓))) ↔ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓))))
8	5, 6, 7	sbimALT 2584	. . . 4 ⊢ (((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓))) ↔ (𝜃 → 𝜏))
9		biid 264	. . . . 5 ⊢ (((𝑥 = 𝑦 → (𝜓 → 𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜓 → 𝜑))) ↔ ((𝑥 = 𝑦 → (𝜓 → 𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜓 → 𝜑))))
10	6, 5, 9	sbimALT 2584	. . . 4 ⊢ (((𝑥 = 𝑦 → (𝜓 → 𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜓 → 𝜑))) ↔ (𝜏 → 𝜃))
11	8, 10	anbi12i 629	. . 3 ⊢ ((((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓))) ∧ ((𝑥 = 𝑦 → (𝜓 → 𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜓 → 𝜑)))) ↔ ((𝜃 → 𝜏) ∧ (𝜏 → 𝜃)))
12	7, 9, 2	sbanALT 2586	. . 3 ⊢ (((𝑥 = 𝑦 → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))) ↔ (((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓))) ∧ ((𝑥 = 𝑦 → (𝜓 → 𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜓 → 𝜑)))))
13		dfbi2 478	. . 3 ⊢ ((𝜃 ↔ 𝜏) ↔ ((𝜃 → 𝜏) ∧ (𝜏 → 𝜃)))
14	11, 12, 13	3bitr4i 306	. 2 ⊢ (((𝑥 = 𝑦 → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))) ↔ (𝜃 ↔ 𝜏))
15	4, 14	bitri 278	1 ⊢ (𝜂 ↔ (𝜃 ↔ 𝜏))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786

This theorem is referenced by:  sblbisALT  2588