Theorem raleleqALT

Description: Alternate proof of raleleq using ralel , being longer and using more axioms. (Contributed by AV, 30-Oct-2020) (New usage is discouraged.) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		ralel	⊢ ∀ 𝑥 ∈ 𝐵 𝑥 ∈ 𝐵
2		id	⊢ ( 𝐴 = 𝐵 → 𝐴 = 𝐵 )
3	2	raleqdv	⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐵 𝑥 ∈ 𝐵 ) )
4	1 3	mpbiri	⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 )