Theorem ralanid 3089

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 534	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
2	1	bicomd 225	. 2 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝜑))
3	2	ralbiia 3085	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 208   ∧ wa 397   ∈ wcel 2121  ∀wral 3055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817

This theorem depends on definitions:  df-bi 209  df-an 398  df-ral 3056

This theorem is referenced by:  idinxpssinxp2  38706