Theorem bianass 643

Description: An inference to merge two lists of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.)

Hypothesis

Ref	Expression
bianass.1	⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))

Assertion

Ref	Expression
bianass	⊢ ((𝜂 ∧ 𝜑) ↔ ((𝜂 ∧ 𝜓) ∧ 𝜒))

Proof of Theorem bianass

Step	Hyp	Ref	Expression
1		bianass.1	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
2	1	anbi2i 624	. 2 ⊢ ((𝜂 ∧ 𝜑) ↔ (𝜂 ∧ (𝜓 ∧ 𝜒)))
3		anass 468	. 2 ⊢ (((𝜂 ∧ 𝜓) ∧ 𝜒) ↔ (𝜂 ∧ (𝜓 ∧ 𝜒)))
4	2, 3	bitr4i 278	1 ⊢ ((𝜂 ∧ 𝜑) ↔ ((𝜂 ∧ 𝜓) ∧ 𝜒))

Syntax hints:   ↔ wb 206   ∧ wa 395

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 207  df-an 396

This theorem is referenced by:  bianassc  644  an12  646  an4  657  cnvresima  6188  elcncf1di  24872  nb3grpr2  29466  dfpth2  29812  wwlksnextwrd  29980  cusgr3cyclex  35334  satfvsuclem2  35558  bj-prmoore  37443  bj-imdirco  37520  redundpim3  39049  isthincd2  49924