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 634	. 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 1542   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 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-v 3399  df-in 3848  df-ss 3858  df-po 5438  df-so 5439  df-fr 5478  df-we 5480

This theorem is referenced by:  ordeq  6173  0we1  8155  oieq2  9043  hartogslem1  9072  wemapwe  9226  ween  9528  dfac8  9628  weth  9988  fpwwe2cbv  10123  fpwwe2lem2  10125  fpwwe2lem4  10127  fpwwecbv  10137  fpwwelem  10138  canthwelem  10143  canthwe  10144  pwfseqlem4a  10154  pwfseqlem4  10155  ltweuz  13413  ltwenn  13414  bpolylem  15487  ltbwe  20848  vitali  24358  weeq12d  40421  aomclem6  40440  omeiunle  43581