Theorem sb6fALT 2602

Description: Alternate version of sb6f 2537. (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.p5	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
sb6fALT.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb6fALT	⊢ (𝜃 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Proof of Theorem sb6fALT

Step	Hyp	Ref	Expression
1		dfsb1.p5	. . . 4 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2		biid 263	. . . 4 ⊢ (((𝑥 = 𝑦 → ∀𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)) ↔ ((𝑥 = 𝑦 → ∀𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)))
3		sb6fALT.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
4	3	nf5ri 2195	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
5	1, 2, 4	sbimiALT 2577	. . 3 ⊢ (𝜃 → ((𝑥 = 𝑦 → ∀𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)))
6	2	sb4aALT 2598	. . 3 ⊢ (((𝑥 = 𝑦 → ∀𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
7	5, 6	syl 17	. 2 ⊢ (𝜃 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
8	1	sb2ALT 2587	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜃)
9	7, 8	impbii 211	1 ⊢ (𝜃 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → 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-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:  sb5fALT  2603