Theorem ordeq 4407

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 4137	. . 3 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2		raleq 2693	. . 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 4402	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
5		dford3 4402	. 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 1364  ∀wral 2475  Tr wtr 4131  Ord word 4397

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-in 3163  df-ss 3170  df-uni 3840  df-tr 4132  df-iord 4401

This theorem is referenced by:  elong  4408  limeq  4412  ordelord  4416  ordtriexmidlem  4555  2ordpr  4560  issmo  6346  issmo2  6347  smoeq  6348  smores  6350  smores2  6352  smodm2  6353  smoiso  6360  tfrlem8  6376  tfri1dALT  6409