Theorem ordeq 4462

Description: Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)

Assertion

Ref	Expression
ordeq	⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))

Proof of Theorem ordeq

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		treq 4187	. . 3 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2		raleq 2728	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 Tr 𝑥 ↔ ∀𝑥 ∈ 𝐵 Tr 𝑥))
3	1, 2	anbi12d 473	. 2 ⊢ (𝐴 = 𝐵 → ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥) ↔ (Tr 𝐵 ∧ ∀𝑥 ∈ 𝐵 Tr 𝑥)))
4		dford3 4457	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
5		dford3 4457	. 2 ⊢ (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥 ∈ 𝐵 Tr 𝑥))
6	3, 4, 5	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ∀wral 2508  Tr wtr 4181  Ord word 4452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-in 3203  df-ss 3210  df-uni 3888  df-tr 4182  df-iord 4456

This theorem is referenced by:  elong  4463  limeq  4467  ordelord  4471  ordtriexmidlem  4610  2ordpr  4615  issmo  6432  issmo2  6433  smoeq  6434  smores  6436  smores2  6438  smodm2  6439  smoiso  6446  tfrlem8  6462  tfri1dALT  6495