Theorem bi3or 607

Description: Join antecedents and consequents with disjunction.

Hypotheses

Ref	Expression
bi3.1	⊢ (φ ↔ ψ)
bi3.2	⊢ (χ ↔ θ)
bi3.3	⊢ (τ ↔ η)

Assertion

Ref	Expression
bi3or	⊢ ((φ ∨ χ ∨ τ) ↔ (ψ ∨ θ ∨ η))

Proof of Theorem bi3or

Step	Hyp	Ref	Expression
1		bi3.1	. . . 4 ⊢ (φ ↔ ψ)
2		bi3.2	. . . 4 ⊢ (χ ↔ θ)
3	1, 2	orbi12i 216	. . 3 ⊢ ((φ ∨ χ) ↔ (ψ ∨ θ))
4		bi3.3	. . 3 ⊢ (τ ↔ η)
5	3, 4	orbi12i 216	. 2 ⊢ (((φ ∨ χ) ∨ τ) ↔ ((ψ ∨ θ) ∨ η))
6		df-3or 582	. 2 ⊢ ((φ ∨ χ ∨ τ) ↔ ((φ ∨ χ) ∨ τ))
7		df-3or 582	. 2 ⊢ ((ψ ∨ θ ∨ η) ↔ ((ψ ∨ θ) ∨ η))
8	5, 6, 7	3bitr4 158	1 ⊢ ((φ ∨ χ ∨ τ) ↔ (ψ ∨ θ ∨ η))

Syntax hints:   ↔ wb 127   ∨ wo 195   ∨ w3o 580

This theorem is referenced by:  wecmpep 2193  ordon 2238  zorn2 3612

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-3or 582

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