Theorem weeq1 5612

Description: Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)

Assertion

Ref	Expression
weeq1	⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴))

Proof of Theorem weeq1

Step	Hyp	Ref	Expression
1		freq1 5592	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴))
2		soeq1 5554	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴))
3	1, 2	anbi12d 638	. 2 ⊢ (𝑅 = 𝑆 → ((𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) ↔ (𝑆 Fr 𝐴 ∧ 𝑆 Or 𝐴)))
4		df-we 5580	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
5		df-we 5580	. 2 ⊢ (𝑆 We 𝐴 ↔ (𝑆 Fr 𝐴 ∧ 𝑆 Or 𝐴))
6	3, 4, 5	3bitr4g 315	1 ⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   Or wor 5532   Fr wfr 5575   We wwe 5577

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 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-ex 1787  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-br 5080  df-po 5533  df-so 5534  df-fr 5578  df-we 5580

This theorem is referenced by:  weeq12d  5614  oieq1  9424  hartogslem1  9454  wemapwe  9616  infxpenlem  9933  dfac8b  9951  ac10ct  9954  canthnumlem  10569  canthp1lem2  10574  pwfseqlem4a  10582  pwfseqlem4  10583  ltbwe  22027  vitali  25605  numiunnum  36705  fin2so  37981  dnwech  43500  aomclem5  43510  aomclem6  43511  aomclem7  43512