Theorem sbbiiALT 2577

Description: Alternate version of sbbii 2080. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dfsb1.ph	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
dfsb1.ps	⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
sbbiiALT.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
sbbiiALT	⊢ (𝜃 ↔ 𝜏)

Proof of Theorem sbbiiALT

Step	Hyp	Ref	Expression
1		dfsb1.ph	. . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2		dfsb1.ps	. . 3 ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
3		sbbiiALT.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
4	3	biimpi 218	. . 3 ⊢ (𝜑 → 𝜓)
5	1, 2, 4	sbimiALT 2576	. 2 ⊢ (𝜃 → 𝜏)
6	3	biimpri 230	. . 3 ⊢ (𝜓 → 𝜑)
7	2, 1, 6	sbimiALT 2576	. 2 ⊢ (𝜏 → 𝜃)
8	5, 7	impbii 211	1 ⊢ (𝜃 ↔ 𝜏)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780

This theorem is referenced by:  sbanALT  2609  sbbiALT  2610  sb7fALT  2615