Theorem weeq2 5626

Description: Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)

Assertion

Ref	Expression
weeq2	⊢ (𝐴 = 𝐵 → (𝑅 We 𝐴 ↔ 𝑅 We 𝐵))

Proof of Theorem weeq2

Step	Hyp	Ref	Expression
1		freq2 5606	. . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵))
2		soeq2 5568	. . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵))
3	1, 2	anbi12d 632	. 2 ⊢ (𝐴 = 𝐵 → ((𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵)))
4		df-we 5593	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
5		df-we 5593	. 2 ⊢ (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵))
6	3, 4, 5	3bitr4g 314	1 ⊢ (𝐴 = 𝐵 → (𝑅 We 𝐴 ↔ 𝑅 We 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   Or wor 5545   Fr wfr 5588   We wwe 5590

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 2008  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2721  df-ral 3045  df-ss 3931  df-po 5546  df-so 5547  df-fr 5591  df-we 5593

This theorem is referenced by:  weeq12d  5627  ordeq  6339  0we1  8470  oieq2  9466  wemapwe  9650  ween  9988  dfac8  10089  weth  10448  pwfseqlem4a  10614  pwfseqlem4  10615  ltweuz  13926  ltwenn  13927  bpolylem  16014  ltbwe  21951  vitali  25514  aomclem6  43048  omeiunle  46515