Theorem 3biant1d 1441

Description: A conjunction is equivalent to a threefold conjunction with single truth, analogous to biantrud 528. (Contributed by Alexander van der Vekens, 26-Sep-2017.)

Hypothesis

Ref	Expression
3biantd.1	⊢ (𝜑 → 𝜃)

Assertion

Ref	Expression
3biant1d	⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒 ∧ 𝜓)))

Proof of Theorem 3biant1d

Step	Hyp	Ref	Expression
1		3biantd.1	. . 3 ⊢ (𝜑 → 𝜃)
2	1	biantrurd 529	. 2 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ (𝜒 ∧ 𝜓))))
3		3anass 1042	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜓) ↔ (𝜃 ∧ (𝜒 ∧ 𝜓)))
4	2, 3	syl6bbr 278	1 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039

This theorem is referenced by:  metuel2  22370  itgsubst  23812  clwlkclwwlk  26903  dfgcd3  33170  itg2addnclem2  33462