Theorem imnang 1842

Description: Quantified implication in terms of quantified negation of conjunction. (Contributed by BJ, 16-Jul-2021.)

Assertion

Ref	Expression
imnang	⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓))

Proof of Theorem imnang

Step	Hyp	Ref	Expression
1		imnan 399	. 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
2	1	albii 1819	1 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by:  alinexa  1843  raln  3069  rexab  3700  n0el  4364  ballotlem2  34491