Theorem weeq1 5625

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 5605	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴))
2		soeq1 5567	. . 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 5593	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
5		df-we 5593	. 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 5545   Fr wfr 5588   We wwe 5590

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 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-ex 1780  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-br 5108  df-po 5546  df-so 5547  df-fr 5591  df-we 5593

This theorem is referenced by:  weeq12d  5627  oieq1  9465  hartogslem1  9495  wemapwe  9650  infxpenlem  9966  dfac8b  9984  ac10ct  9987  canthnumlem  10601  canthp1lem2  10606  pwfseqlem4a  10614  pwfseqlem4  10615  ltbwe  21951  vitali  25514  numiunnum  36458  fin2so  37601  dnwech  43037  aomclem5  43047  aomclem6  43048  aomclem7  43049