Theorem raleleq 3335

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 3083	. 2 ⊢ ∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐵
2		raleq 3320	. 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐵))
3	1, 2	mpbiri 261	1 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∀wral 3080

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-ral 3081  df-rex 3091

This theorem is used by:  uvtxnbgrb  30039