Theorem orbi1i 764

Description: Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
orbi2i.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
orbi1i	⊢ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒))

Proof of Theorem orbi1i

Step	Hyp	Ref	Expression
1		orcom 729	. 2 ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜒 ∨ 𝜑))
2		orbi2i.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	orbi2i 763	. 2 ⊢ ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓))
4		orcom 729	. 2 ⊢ ((𝜒 ∨ 𝜓) ↔ (𝜓 ∨ 𝜒))
5	1, 3, 4	3bitri 206	1 ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒))

Syntax hints:   ↔ wb 105   ∨ wo 709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  orbi12i  765  orordi  774  3or6  1334  19.45  1694  sbequilem  1849  unass  3307  frecsuc  6426  nninfwlporlemd  7188  elznn0nn  9285