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Theorem raleleqALT 3134
 Description: Alternate proof of raleleq 3133 using ralel 2907, 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 → ∀𝑥𝐴 𝑥𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem raleleqALT
StepHypRef Expression
1 ralel 2907 . 2 𝑥𝐵 𝑥𝐵
2 id 22 . . 3 (𝐴 = 𝐵𝐴 = 𝐵)
32raleqdv 3121 . 2 (𝐴 = 𝐵 → (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥𝐵 𝑥𝐵))
41, 3mpbiri 247 1 (𝐴 = 𝐵 → ∀𝑥𝐴 𝑥𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901 This theorem is referenced by: (None)
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