Theorem sb2ALT 2587

Description: Alternate version of sb2 2504. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
sb2ALT	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜃)

Proof of Theorem sb2ALT

Step	Hyp	Ref	Expression
1		sp 2182	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
2		equs4 2438	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
3		dfsb1.ph	. 2 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
4	1, 2, 3	sylanbrc 585	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜃)

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-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by:  stdpc4ALT  2590  dfsb2ALT  2591  sbequiALT  2596  hbsb2ALT  2599  equsb1ALT  2601  sb6fALT  2602  sbi1ALT  2606