Theorem raleleq 3156

Description: All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020.)

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		eleq2 2690	. . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	biimpd 219	. 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
3	2	ralrimiv 2965	1 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ∀wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-ral 2917

This theorem is referenced by:  uvtxnbgrb  26302  cplgruvtxb  26311