Theorem weeq12d 5652

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 5650	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴))
4		weeq2 5651	. . 3 ⊢ (𝐴 = 𝐵 → (𝑆 We 𝐴 ↔ 𝑆 We 𝐵))
5	3, 4	sylan9bb 518	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
6	1, 2, 5	syl2anc 595	1 ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   We wwe 5615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-ex 1810  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-ss 3923  df-br 5111  df-po 5571  df-so 5572  df-fr 5616  df-we 5618

This theorem is referenced by:  hartogslem1  9505  fpwwe2cbv  10616  fpwwe2lem2  10618  fpwwe2lem4  10620  fpwwecbv  10630  fpwwelem  10631  canthwelem  10636  canthwe  10637  pwfseqlem4  10648  fnwe2lem1  43757  aomclem1  43761  aomclem4  43764  aomclem5  43765  aomclem6  43766