Theorem hbra2VD 41192

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::	⊢ (∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑 → ∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
2::	⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔ ∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
3:1,2,?: e00 41100	⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 → ∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
4:2:	⊢ ∀𝑦(∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔ ∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
5:4,?: e0a 41104	⊢ (∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔ ∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
qed:3,5,?: e00 41100	⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 → ∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑)

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
hbra2VD	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)

Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem hbra2VD

Step	Hyp	Ref	Expression
1		ralcom 3354	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)
2		hbra1 3220	. 2 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑 → ∀𝑦∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)
3	1, 2	hbxfrbi 1821	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)

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)