Theorem hbra2VD 44880

Description: Virtual deduction proof of nfra2 3376. 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 44788	⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 → ∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
4:2:	⊢ ∀𝑦(∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔ ∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
5:4,?: e0a 44792	⊢ (∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵𝜑 ↔ ∀𝑦∀𝑦 ∈ 𝐵∀𝑥 ∈ 𝐴𝜑)
qed:3,5,?: e00 44788	⊢ (∀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 3289	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)
2		hbra1 3301	. 2 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑 → ∀𝑦∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)
3	1, 2	hbxfrbi 1825	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4  ∀wal 1538  ∀wral 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-ral 3062

This theorem is referenced by: (None)