Theorem weeq2 5609

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 5589	. . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵))
2		soeq2 5551	. . 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 5576	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
5		df-we 5576	. 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 1541   Or wor 5528   Fr wfr 5571   We wwe 5573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2725  df-ral 3049  df-ss 3915  df-po 5529  df-so 5530  df-fr 5574  df-we 5576

This theorem is referenced by:  weeq12d  5610  ordeq  6321  0we1  8430  oieq2  9410  wemapwe  9598  ween  9937  dfac8  10038  weth  10397  pwfseqlem4a  10563  pwfseqlem4  10564  ltweuz  13875  ltwenn  13876  bpolylem  15962  ltbwe  21990  vitali  25561  aomclem6  43216  omeiunle  46677