Theorem bj-notalbii

Description: Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 (103>94), ballotlem2 (2655>2648), bnj1143 (522>519), hausdiag (2119>2104). (Contributed by BJ, 17-Jul-2021)

	Ref	Expression
Hypothesis	bj-notalbii.1	⊢ ( 𝜑 ↔ 𝜓 )
Assertion	bj-notalbii	⊢ ( ∀ 𝑥 ¬ 𝜑 ↔ ∀ 𝑥 ¬ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bj-notalbii.1	⊢ ( 𝜑 ↔ 𝜓 )
2	1	notbii	⊢ ( ¬ 𝜑 ↔ ¬ 𝜓 )
3	2	albii	⊢ ( ∀ 𝑥 ¬ 𝜑 ↔ ∀ 𝑥 ¬ 𝜓 )