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Theorem isoeq3 5386
 Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
Assertion
Ref Expression
isoeq3 (𝑆 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑇 (A, B)))

Proof of Theorem isoeq3
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 3757 . . . . 5 (𝑆 = 𝑇 → ((𝐻x)𝑆(𝐻y) ↔ (𝐻x)𝑇(𝐻y)))
21bibi2d 221 . . . 4 (𝑆 = 𝑇 → ((x𝑅y ↔ (𝐻x)𝑆(𝐻y)) ↔ (x𝑅y ↔ (𝐻x)𝑇(𝐻y))))
322ralbidv 2342 . . 3 (𝑆 = 𝑇 → (x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y)) ↔ x A y A (x𝑅y ↔ (𝐻x)𝑇(𝐻y))))
43anbi2d 437 . 2 (𝑆 = 𝑇 → ((𝐻:A1-1-ontoB x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y))) ↔ (𝐻:A1-1-ontoB x A y A (x𝑅y ↔ (𝐻x)𝑇(𝐻y)))))
5 df-isom 4854 . 2 (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ (𝐻:A1-1-ontoB x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
6 df-isom 4854 . 2 (𝐻 Isom 𝑅, 𝑇 (A, B) ↔ (𝐻:A1-1-ontoB x A y A (x𝑅y ↔ (𝐻x)𝑇(𝐻y))))
74, 5, 63bitr4g 212 1 (𝑆 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑇 (A, B)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∀wral 2300   class class class wbr 3755  –1-1-onto→wf1o 4844  ‘cfv 4845   Isom wiso 4846 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033  df-ral 2305  df-br 3756  df-isom 4854 This theorem is referenced by: (None)
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