This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem bj-nnfbid

Description: Nonfreeness in both sides implies nonfreeness in the biconditional, deduction form. (Contributed by BJ, 2-Dec-2023) (Proof modification is discouraged.)

Ref Expression
Hypotheses bj-nnfbid.1 
|- ( ph -> F// x ps )
bj-nnfbid.2 
|- ( ph -> F// x ch )
Assertion bj-nnfbid 
|- ( ph -> F// x ( ps <-> ch ) )

Proof

Step Hyp Ref Expression
1 bj-nnfbid.1 
 |-  ( ph -> F// x ps )
2 bj-nnfbid.2 
 |-  ( ph -> F// x ch )
3 bj-nnfim 
 |-  ( ( F// x ps /\ F// x ch ) -> F// x ( ps -> ch ) )
4 1 2 3 syl2anc 
 |-  ( ph -> F// x ( ps -> ch ) )
5 bj-nnfim 
 |-  ( ( F// x ch /\ F// x ps ) -> F// x ( ch -> ps ) )
6 2 1 5 syl2anc 
 |-  ( ph -> F// x ( ch -> ps ) )
7 4 6 bj-nnfand 
 |-  ( ph -> F// x ( ( ps -> ch ) /\ ( ch -> ps ) ) )
8 dfbi2 
 |-  ( ( ps <-> ch ) <-> ( ( ps -> ch ) /\ ( ch -> ps ) ) )
9 8 bj-nnfbii 
 |-  ( F// x ( ps <-> ch ) <-> F// x ( ( ps -> ch ) /\ ( ch -> ps ) ) )
10 7 9 sylibr 
 |-  ( ph -> F// x ( ps <-> ch ) )