Theorem raleleqOLD 3318

Description: Obsolete version of raleleq 3317 as of 18-Jul-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		ralel 3048	. 2 ⊢ ∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐵
2		id 22	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
3	2	raleqdv 3301	. 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐵))
4	1, 3	mpbiri 258	1 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2722  df-ral 3046  df-rex 3055

This theorem is referenced by: (None)