Theorem 3biant1d 1474

Description: A conjunction is equivalent to a threefold conjunction with single truth, analogous to biantrud 534. (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 535	. 2 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ (𝜒 ∧ 𝜓))))
3		3anass 1091	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜓) ↔ (𝜃 ∧ (𝜒 ∧ 𝜓)))
4	2, 3	syl6bbr 291	1 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083

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 209  df-an 399  df-3an 1085

This theorem is referenced by:  metuel2  23169  itgsubst  24640  clwlkclwwlk  27774  dfgcd3  34599  itg2addnclem2  34938