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Theorem raleleqALT 3427
 Description: Alternate proof of raleleq 3426 using ralel 3147, 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 3147 . 2 𝑥𝐵 𝑥𝐵
2 id 22 . . 3 (𝐴 = 𝐵𝐴 = 𝐵)
32raleqdv 3414 . 2 (𝐴 = 𝐵 → (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥𝐵 𝑥𝐵))
41, 3mpbiri 260 1 (𝐴 = 𝐵 → ∀𝑥𝐴 𝑥𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1531   ∈ wcel 2108  ∀wral 3136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1775  df-cleq 2812  df-clel 2891  df-ral 3141 This theorem is referenced by: (None)
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