Theorem sbequ2ALT 2556

Description: Alternate version of sbequ2 2247. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-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
sbequ2ALT	⊢ (𝑥 = 𝑦 → (𝜃 → 𝜑))

Proof of Theorem sbequ2ALT

Step	Hyp	Ref	Expression
1		dfsb1.ph	. . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2	1	simplbi 501	. 2 ⊢ (𝜃 → (𝑥 = 𝑦 → 𝜑))
3	2	com12 32	1 ⊢ (𝑥 = 𝑦 → (𝜃 → 𝜑))

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

This theorem depends on definitions:  df-bi 210  df-an 400

This theorem is referenced by:  sbequ12ALT  2557  dfsb2ALT  2567  sbequiALT  2572  sbi1ALT  2582