Theorem weeq12d 38562

Description: Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Hypotheses

Ref	Expression
weeq12d.l	⊢ (𝜑 → 𝑅 = 𝑆)
weeq12d.r	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
weeq12d	⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))

Proof of Theorem weeq12d

Step	Hyp	Ref	Expression
1		weeq12d.l	. . 3 ⊢ (𝜑 → 𝑅 = 𝑆)
2		weeq1 5343	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴))
4		weeq12d.r	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
5		weeq2 5344	. . 3 ⊢ (𝐴 = 𝐵 → (𝑆 We 𝐴 ↔ 𝑆 We 𝐵))
6	4, 5	syl 17	. 2 ⊢ (𝜑 → (𝑆 We 𝐴 ↔ 𝑆 We 𝐵))
7	3, 6	bitrd 271	1 ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1601   We wwe 5313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-ral 3094  df-rex 3095  df-in 3798  df-ss 3805  df-br 4887  df-po 5274  df-so 5275  df-fr 5314  df-we 5316

This theorem is referenced by:  fnwe2lem1  38572  aomclem1  38576  aomclem4  38579  aomclem5  38580  aomclem6  38581