Theorem weeq12d 5678

Description: Equality deduction for well-orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.) (Proof shortened by Matthew House, 10-Sep-2025.)

Hypotheses

Ref	Expression
weeq12d.1	⊢ (𝜑 → 𝑅 = 𝑆)
weeq12d.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

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

Proof of Theorem weeq12d

Step	Hyp	Ref	Expression
1		weeq12d.1	. 2 ⊢ (𝜑 → 𝑅 = 𝑆)
2		weeq12d.2	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		weeq1 5676	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴))
4		weeq2 5677	. . 3 ⊢ (𝐴 = 𝐵 → (𝑆 We 𝐴 ↔ 𝑆 We 𝐵))
5	3, 4	sylan9bb 509	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
6	1, 2, 5	syl2anc 584	1 ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   We wwe 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-ex 1777  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-ss 3980  df-br 5149  df-po 5597  df-so 5598  df-fr 5641  df-we 5643

This theorem is referenced by:  hartogslem1  9580  fpwwe2cbv  10668  fpwwe2lem2  10670  fpwwe2lem4  10672  fpwwecbv  10682  fpwwelem  10683  canthwelem  10688  canthwe  10689  pwfseqlem4  10700  fnwe2lem1  43039  aomclem1  43043  aomclem4  43046  aomclem5  43047  aomclem6  43048