Theorem weeq12d 40860

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 5577	. . 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 5578	. . 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 278	1 ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542   We wwe 5543

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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-v 3433  df-in 3899  df-ss 3909  df-br 5080  df-po 5503  df-so 5504  df-fr 5544  df-we 5546

This theorem is referenced by:  fnwe2lem1  40870  aomclem1  40874  aomclem4  40877  aomclem5  40878  aomclem6  40879