Theorem sbequ1ALT 2555

Description: Alternate version of sbequ1 2246. (Contributed by NM, 16-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
sbequ1ALT	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜃))

Proof of Theorem sbequ1ALT

Step	Hyp	Ref	Expression
1		pm3.4 809	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜑))
2		19.8a 2178	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
3		dfsb1.ph	. . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
4	1, 2, 3	sylanbrc 586	. 2 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜃)
5	4	ex 416	1 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜃))

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-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782

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