Theorem 3orbi123 45248

Description: pm4.39 991 with a 3-conjunct antecedent. This proof is 3orbi123VD 45586 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
3orbi123	⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂)))

Proof of Theorem 3orbi123

Step	Hyp	Ref	Expression
1		simp1 1153	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (𝜑 ↔ 𝜓))
2		simp2 1154	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (𝜒 ↔ 𝜃))
3		simp3 1155	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (𝜏 ↔ 𝜂))
4	1, 2, 3	3orbi123d 1462	1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ w3o 1101   ∧ w3a 1102

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104

This theorem is used by:  sbcoreleleq  45272  sbcoreleleqVD  45595