Theorem weeq1 5633

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 5613	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴))
2		soeq1 5575	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴))
3	1, 2	anbi12d 632	. 2 ⊢ (𝑅 = 𝑆 → ((𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) ↔ (𝑆 Fr 𝐴 ∧ 𝑆 Or 𝐴)))
4		df-we 5601	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
5		df-we 5601	. 2 ⊢ (𝑆 We 𝐴 ↔ (𝑆 Fr 𝐴 ∧ 𝑆 Or 𝐴))
6	3, 4, 5	3bitr4g 314	1 ⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   Or wor 5553   Fr wfr 5596   We wwe 5598

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-ex 1780  df-cleq 2722  df-clel 2804  df-ral 3047  df-rex 3056  df-br 5116  df-po 5554  df-so 5555  df-fr 5599  df-we 5601

This theorem is referenced by:  weeq12d  5635  oieq1  9483  hartogslem1  9513  wemapwe  9668  infxpenlem  9984  dfac8b  10002  ac10ct  10005  canthnumlem  10619  canthp1lem2  10624  pwfseqlem4a  10632  pwfseqlem4  10633  ltbwe  21957  vitali  25521  numiunnum  36455  fin2so  37598  dnwech  43009  aomclem5  43019  aomclem6  43020  aomclem7  43021