Theorem ordeq 4374

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 4109	. . 3 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2		raleq 2673	. . 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 4369	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
5		dford3 4369	. 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 1353  ∀wral 2455  Tr wtr 4103  Ord word 4364

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-in 3137  df-ss 3144  df-uni 3812  df-tr 4104  df-iord 4368

This theorem is referenced by:  elong  4375  limeq  4379  ordelord  4383  ordtriexmidlem  4520  2ordpr  4525  issmo  6291  issmo2  6292  smoeq  6293  smores  6295  smores2  6297  smodm2  6298  smoiso  6305  tfrlem8  6321  tfri1dALT  6354