Theorem weeq12d 39647

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 5545	. . 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 5546	. . 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 281	1 ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   We wwe 5515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-ral 3145  df-rex 3146  df-in 3945  df-ss 3954  df-br 5069  df-po 5476  df-so 5477  df-fr 5516  df-we 5518

This theorem is referenced by:  fnwe2lem1  39657  aomclem1  39661  aomclem4  39664  aomclem5  39665  aomclem6  39666