Theorem orbi1d 467

Description: Deduction adding a right disjunct to both sides of a logical equivalence.

Hypothesis

Ref	Expression
bid.1	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
orbi1d	⊢ (φ → ((ψ ∨ θ) ↔ (χ ∨ θ)))

Proof of Theorem orbi1d

Step	Hyp	Ref	Expression
1		bid.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2	1	orbi2d 466	. 2 ⊢ (φ → ((θ ∨ ψ) ↔ (θ ∨ χ)))
3		orcom 209	. 2 ⊢ ((ψ ∨ θ) ↔ (θ ∨ ψ))
4		orcom 209	. 2 ⊢ ((χ ∨ θ) ↔ (θ ∨ χ))
5	2, 3, 4	3bitr4g 428	1 ⊢ (φ → ((ψ ∨ θ) ↔ (χ ∨ θ)))

Syntax hints:   → wi 2   ↔ wb 127   ∨ wo 195

This theorem is referenced by:  orbi12d 475  dedlema 569  eueq2 1429  eueq3 1430  uneq1 1605  r19.45zv 1770  ifeq1 1778  mul0ort 4212  h1datomt 5484

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-an 198

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