Theorem ralanid 3094

Description: Cancellation law for restricted universal quantification. (Contributed by Peter Mazsa, 30-Dec-2018.) (Proof shortened by Wolf Lammen, 29-Jun-2023.)

Assertion

Ref	Expression
ralanid	⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑)

Proof of Theorem ralanid

Step	Hyp	Ref	Expression
1		ibar 528	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
2	1	bicomd 223	. 2 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝜑))
3	2	ralbiia 3090	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2107  ∀wral 3060

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  df-an 396  df-ral 3061

This theorem is referenced by:  idinxpssinxp2  38320