Theorem weeq2 5651

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 5631	. . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵))
2		soeq2 5593	. . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵))
3	1, 2	anbi12d 644	. 2 ⊢ (𝐴 = 𝐵 → ((𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵)))
4		df-we 5618	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
5		df-we 5618	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   Or wor 5570   Fr wfr 5613   We wwe 5615

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2757  df-ral 3082  df-ss 3923  df-po 5571  df-so 5572  df-fr 5616  df-we 5618

This theorem is used by:  weeq12d  5652  ordeq  6371  0we1  8493  oieq2  9478  wemapwe  9669  ween  10031  dfac8  10131  weth  10490  pwfseqlem4a  10657  pwfseqlem4  10658  ltweuz  14011  ltwenn  14012  bpolylem  16120  ltbwe  22225  vitali  25803  aomclem6  43819  omeiunle  47264