Theorem weeq12d 5689

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 5687	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴))
4		weeq2 5688	. . 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 583	1 ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   We wwe 5651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-ex 1778  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-ss 3993  df-br 5167  df-po 5607  df-so 5608  df-fr 5652  df-we 5654

This theorem is referenced by:  hartogslem1  9611  fpwwe2cbv  10699  fpwwe2lem2  10701  fpwwe2lem4  10703  fpwwecbv  10713  fpwwelem  10714  canthwelem  10719  canthwe  10720  pwfseqlem4  10731  fnwe2lem1  43007  aomclem1  43011  aomclem4  43014  aomclem5  43015  aomclem6  43016