Theorem raleleqOLD 3334

Description: Obsolete version of raleleq 3331 as of 9-Mar-2025. (Contributed by AV, 30-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem raleleqOLD

Step	Hyp	Ref	Expression
1		eleq2 2816	. . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	biimpd 228	. 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
3	2	ralrimiv 3139	1 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ∀wral 3055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-cleq 2718  df-clel 2804  df-ral 3056

This theorem is referenced by: (None)