Theorem sbequALT 2597

Description: Alternate version of sbequ 2090. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dfsb1.xz	⊢ (𝜃 ↔ ((𝑧 = 𝑥 → 𝜑) ∧ ∃𝑧(𝑧 = 𝑥 ∧ 𝜑)))
dfsb1.yz	⊢ (𝜏 ↔ ((𝑧 = 𝑦 → 𝜑) ∧ ∃𝑧(𝑧 = 𝑦 ∧ 𝜑)))

Assertion

Ref	Expression
sbequALT	⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜏))

Proof of Theorem sbequALT

Step	Hyp	Ref	Expression
1		dfsb1.xz	. . 3 ⊢ (𝜃 ↔ ((𝑧 = 𝑥 → 𝜑) ∧ ∃𝑧(𝑧 = 𝑥 ∧ 𝜑)))
2		dfsb1.yz	. . 3 ⊢ (𝜏 ↔ ((𝑧 = 𝑦 → 𝜑) ∧ ∃𝑧(𝑧 = 𝑦 ∧ 𝜑)))
3	1, 2	sbequiALT 2596	. 2 ⊢ (𝑥 = 𝑦 → (𝜃 → 𝜏))
4	2, 1	sbequiALT 2596	. . 3 ⊢ (𝑦 = 𝑥 → (𝜏 → 𝜃))
5	4	equcoms 2027	. 2 ⊢ (𝑥 = 𝑦 → (𝜏 → 𝜃))
6	3, 5	impbid 214	1 ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜏))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∃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  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:  sbco2ALT  2615