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Theorem hbra2VD 41194
Description: Virtual deduction proof of nfra2 3228. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1:: (∀𝑦𝐵𝑥𝐴𝜑 𝑦𝑦𝐵𝑥𝐴𝜑)
2:: (∀𝑥𝐴𝑦𝐵𝜑 𝑦𝐵𝑥𝐴𝜑)
3:1,2,?: e00 41102 (∀𝑥𝐴𝑦𝐵𝜑 𝑦𝑦𝐵𝑥𝐴𝜑)
4:2: 𝑦(∀𝑥𝐴𝑦𝐵𝜑 𝑦𝐵𝑥𝐴𝜑)
5:4,?: e0a 41106 (∀𝑦𝑥𝐴𝑦𝐵𝜑 𝑦𝑦𝐵𝑥𝐴𝜑)
qed:3,5,?: e00 41102 (∀𝑥𝐴𝑦𝐵𝜑 𝑦𝑥𝐴𝑦𝐵𝜑)
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
hbra2VD (∀𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝑥𝐴𝑦𝐵 𝜑)
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem hbra2VD
StepHypRef Expression
1 ralcom 3354 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)
2 hbra1 3220 . 2 (∀𝑦𝐵𝑥𝐴 𝜑 → ∀𝑦𝑦𝐵𝑥𝐴 𝜑)
31, 2hbxfrbi 1821 1 (∀𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝑥𝐴𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  wral 3138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2157  ax-12 2173
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-ral 3143
This theorem is referenced by: (None)
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