Theorem weeq12d 5605

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 5603	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴))
4		weeq2 5604	. . 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 1541   We wwe 5568

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-ex 1781  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-ss 3919  df-br 5092  df-po 5524  df-so 5525  df-fr 5569  df-we 5571

This theorem is referenced by:  hartogslem1  9428  fpwwe2cbv  10518  fpwwe2lem2  10520  fpwwe2lem4  10522  fpwwecbv  10532  fpwwelem  10533  canthwelem  10538  canthwe  10539  pwfseqlem4  10550  fnwe2lem1  43082  aomclem1  43086  aomclem4  43089  aomclem5  43090  aomclem6  43091