Theorem weeq1 5665

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 5647	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴))
2		soeq1 5610	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴))
3	1, 2	anbi12d 629	. 2 ⊢ (𝑅 = 𝑆 → ((𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) ↔ (𝑆 Fr 𝐴 ∧ 𝑆 Or 𝐴)))
4		df-we 5634	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
5		df-we 5634	. 2 ⊢ (𝑆 We 𝐴 ↔ (𝑆 Fr 𝐴 ∧ 𝑆 Or 𝐴))
6	3, 4, 5	3bitr4g 313	1 ⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   Or wor 5588   Fr wfr 5629   We wwe 5631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-ex 1780  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-br 5150  df-po 5589  df-so 5590  df-fr 5632  df-we 5634

This theorem is referenced by:  oieq1  9511  hartogslem1  9541  wemapwe  9696  infxpenlem  10012  dfac8b  10030  ac10ct  10033  fpwwe2cbv  10629  fpwwe2lem2  10631  fpwwe2lem4  10633  fpwwecbv  10643  fpwwelem  10644  canthnumlem  10647  canthwelem  10649  canthwe  10650  canthp1lem2  10652  pwfseqlem4a  10660  pwfseqlem4  10661  ltbwe  21820  vitali  25364  fin2so  36780  weeq12d  42086  dnwech  42094  aomclem5  42104  aomclem6  42105  aomclem7  42106