Theorem sblbisALT 2611

Description: Alternate version of sblbis 2318. (Contributed by NM, 19-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dfsb1.p6	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
dfsb1.s4	⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
dfsb1.bi	⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ↔ 𝜓))))
sblbisALT.1	⊢ (𝜏 ↔ 𝜒)

Assertion

Ref	Expression
sblbisALT	⊢ (𝜂 ↔ (𝜃 ↔ 𝜒))

Proof of Theorem sblbisALT

Step	Hyp	Ref	Expression
1		dfsb1.p6	. . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2		dfsb1.s4	. . 3 ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
3		dfsb1.bi	. . 3 ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ↔ 𝜓))))
4	1, 2, 3	sbbiALT 2610	. 2 ⊢ (𝜂 ↔ (𝜃 ↔ 𝜏))
5		sblbisALT.1	. . 3 ⊢ (𝜏 ↔ 𝜒)
6	5	bibi2i 340	. 2 ⊢ ((𝜃 ↔ 𝜏) ↔ (𝜃 ↔ 𝜒))
7	4, 6	bitri 277	1 ⊢ (𝜂 ↔ (𝜃 ↔ 𝜒))

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  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784

This theorem is referenced by:  sbieALT  2612