Metamath Proof Explorer


Theorem bj-nnfbd

Description: If two formulas are equivalent for all x , then nonfreeness of x in one of them is equivalent to nonfreeness in the other, deduction form. See bj-nnfbi . (Contributed by BJ, 27-Aug-2023)

Ref Expression
Hypothesis bj-nnfbd.1 φψχ
Assertion bj-nnfbd φℲ'xψℲ'xχ

Proof

Step Hyp Ref Expression
1 bj-nnfbd.1 φψχ
2 1 alrimiv φxψχ
3 bj-nnfbi ψχxψχℲ'xψℲ'xχ
4 1 2 3 syl2anc φℲ'xψℲ'xχ