Theorem raleleq 3332

Description: All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020.) (Proof shortened by Wolf Lammen, 18-Jul-2025.)

Assertion

Ref	Expression
raleleq	⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem raleleq

Step	Hyp	Ref	Expression
1		ralel 3079	. 2 ⊢ ∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐵
2		raleq 3317	. 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐵))
3	1, 2	mpbiri 260	1 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142  ∀wral 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-cleq 2754  df-ral 3077  df-rex 3087

This theorem is referenced by:  uvtxnbgrb  29602