Theorem raleleqALT 3373

Description: Alternate proof of raleleq 3372 using ralel 3117, being longer and using more axioms. (Contributed by AV, 30-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem raleleqALT

Step	Hyp	Ref	Expression
1		ralel 3117	. 2 ⊢ ∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐵
2		id 22	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
3	2	raleqdv 3364	. 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐵))
4	1, 3	mpbiri 261	1 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ∀wral 3106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870  df-ral 3111

This theorem is referenced by: (None)