Theorem bj-notalbii 36574

Description: Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 4360 (103>94), ballotlem2 34450 (2655>2648), bnj1143 34763 (522>519), hausdiag 23599 (2119>2104). (Contributed by BJ, 17-Jul-2021.)

Hypothesis

Ref	Expression
bj-notalbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bj-notalbii	⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓)

Proof of Theorem bj-notalbii

Step	Hyp	Ref	Expression
1		bj-notalbii.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	notbii 320	. 2 ⊢ (¬ 𝜑 ↔ ¬ 𝜓)
3	2	albii 1818	1 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808

This theorem depends on definitions:  df-bi 207

This theorem is referenced by: (None)