Theorem equncomiVD 45316

Description: Inference form of equncom 4100. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. equncomi 4101 is equncomiVD 45316 without virtual deductions and was automatically derived from equncomiVD 45316.

h1::	⊢ 𝐴 = (𝐵 ∪ 𝐶)
qed:1:	⊢ 𝐴 = (𝐶 ∪ 𝐵)

(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
equncomiVD.1	⊢ 𝐴 = (𝐵 ∪ 𝐶)

Assertion

Ref	Expression
equncomiVD	⊢ 𝐴 = (𝐶 ∪ 𝐵)

Proof of Theorem equncomiVD

Step	Hyp	Ref	Expression
1		equncomiVD.1	. 2 ⊢ 𝐴 = (𝐵 ∪ 𝐶)
2		equncom 4100	. . 3 ⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))
3	2	biimpi 216	. 2 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → 𝐴 = (𝐶 ∪ 𝐵))
4	1, 3	e0a 45219	1 ⊢ 𝐴 = (𝐶 ∪ 𝐵)

Syntax hints:   = wceq 1542   ∪ cun 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-un 3895

This theorem is referenced by: (None)