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Theorem raleleqALT 3305
Description: Alternate proof of raleleq 3304 using ralel 3071, 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 3071 . 2 𝑥𝐵 𝑥𝐵
2 id 22 . . 3 (𝐴 = 𝐵𝐴 = 𝐵)
32raleqdv 3292 . 2 (𝐴 = 𝐵 → (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥𝐵 𝑥𝐵))
41, 3mpbiri 248 1 (𝐴 = 𝐵 → ∀𝑥𝐴 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  wral 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065
This theorem is referenced by: (None)
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