Theorem weeq2 5508

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 5490	. . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵))
2		soeq2 5459	. . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵))
3	1, 2	anbi12d 633	. 2 ⊢ (𝐴 = 𝐵 → ((𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵)))
4		df-we 5480	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
5		df-we 5480	. 2 ⊢ (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵))
6	3, 4, 5	3bitr4g 317	1 ⊢ (𝐴 = 𝐵 → (𝑅 We 𝐴 ↔ 𝑅 We 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   Or wor 5437   Fr wfr 5475   We wwe 5477

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-in 3888  df-ss 3898  df-po 5438  df-so 5439  df-fr 5478  df-we 5480

This theorem is referenced by:  ordeq  6166  0we1  8114  oieq2  8961  hartogslem1  8990  wemapwe  9144  ween  9446  dfac8  9546  weth  9906  fpwwe2cbv  10041  fpwwe2lem2  10043  fpwwe2lem5  10045  fpwwecbv  10055  fpwwelem  10056  canthwelem  10061  canthwe  10062  pwfseqlem4a  10072  pwfseqlem4  10073  ltweuz  13324  ltwenn  13325  bpolylem  15394  ltbwe  20712  vitali  24217  weeq12d  39984  aomclem6  40003  omeiunle  43156