Theorem weeq12d 5655

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 5653	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴))
4		weeq2 5654	. . 3 ⊢ (𝐴 = 𝐵 → (𝑆 We 𝐴 ↔ 𝑆 We 𝐵))
5	3, 4	sylan9bb 519	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
6	1, 2, 5	syl2anc 596	1 ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   We wwe 5618

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-ex 1813  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-ss 3925  df-br 5115  df-po 5574  df-so 5575  df-fr 5619  df-we 5621

This theorem is used by:  hartogslem1  9514  fpwwe2cbv  10633  fpwwe2lem2  10635  fpwwe2lem4  10637  fpwwecbv  10647  fpwwelem  10648  canthwelem  10653  canthwe  10654  pwfseqlem4  10665  fnwe2lem1  43818  aomclem1  43822  aomclem4  43825  aomclem5  43826  aomclem6  43827