Theorem raleleqALT 3348

Description: Alternate proof of raleleq 3347 using ralel 3074, 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 3074	. 2 ⊢ ∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐵
2		id 22	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
3	2	raleqdv 3339	. 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐵))
4	1, 3	mpbiri 257	1 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ∀wral 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-cleq 2730  df-ral 3068

This theorem is referenced by: (None)