Theorem bianassc 467

Description: An inference to merge two lists of conjuncts. (Contributed by Peter Mazsa, 24-Sep-2022.)

Hypothesis

Ref	Expression
bianass.1	|- ( ph <-> ( ps /\ ch ) )

Assertion

Ref	Expression
bianassc	|- ( ( et /\ ph ) <-> ( ( ps /\ et ) /\ ch ) )

Proof of Theorem bianassc

Step	Hyp	Ref	Expression
1		bianass.1	. . 3 |- ( ph <-> ( ps /\ ch ) )
2	1	bianass 466	. 2 |- ( ( et /\ ph ) <-> ( ( et /\ ps ) /\ ch ) )
3		ancom 264	. . 3 |- ( ( et /\ ps ) <-> ( ps /\ et ) )
4	3	anbi1i 455	. 2 |- ( ( ( et /\ ps ) /\ ch ) <-> ( ( ps /\ et ) /\ ch ) )
5	2, 4	bitri 183	1 |- ( ( et /\ ph ) <-> ( ( ps /\ et ) /\ ch ) )

Syntax hints:   /\ wa 103   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  an21  468