Theorem sb6ALT 2585

Description: Alternate version of sb6 2093. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
dfsb1.ph	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

Assertion

Ref	Expression
sb6ALT	⊢ (𝜃 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜃(𝑥,𝑦)

Proof of Theorem sb6ALT

Step	Hyp	Ref	Expression
1		dfsb1.ph	. . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2	1	sb4vOLDALT 2584	. 2 ⊢ (𝜃 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1	sb2vOLDALT 2583	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜃)
4	2, 3	impbii 211	1 ⊢ (𝜃 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃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

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by:  sb5ALT2  2586