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Theorem raleleqALT 3426
Description: Alternate proof of raleleq 3425 using ralel 3146, 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 3146 . 2 𝑥𝐵 𝑥𝐵
2 id 22 . . 3 (𝐴 = 𝐵𝐴 = 𝐵)
32raleqdv 3413 . 2 (𝐴 = 𝐵 → (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥𝐵 𝑥𝐵))
41, 3mpbiri 259 1 (𝐴 = 𝐵 → ∀𝑥𝐴 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wral 3135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-cleq 2811  df-clel 2890  df-ral 3140
This theorem is referenced by: (None)
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