Theorem weeq12d 39984

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 5507	. . 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 5508	. . 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 282	1 ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   We wwe 5477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-v 3443  df-in 3888  df-ss 3898  df-br 5031  df-po 5438  df-so 5439  df-fr 5478  df-we 5480

This theorem is referenced by:  fnwe2lem1  39994  aomclem1  39998  aomclem4  40001  aomclem5  40002  aomclem6  40003