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Theorem isoeq5 5486
 Description: Equality theorem for isomorphisms. (Contributed by set.mm contributors, 17-May-2004.)
Assertion
Ref Expression
isoeq5 (B = C → (H Isom R, S (A, B) ↔ H Isom R, S (A, C)))

Proof of Theorem isoeq5
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oeq3 5283 . . 3 (B = C → (H:A1-1-ontoBH:A1-1-ontoC))
21anbi1d 685 . 2 (B = C → ((H:A1-1-ontoB x A y A (xRy ↔ (Hx)S(Hy))) ↔ (H:A1-1-ontoC x A y A (xRy ↔ (Hx)S(Hy)))))
3 df-iso 4796 . 2 (H Isom R, S (A, B) ↔ (H:A1-1-ontoB x A y A (xRy ↔ (Hx)S(Hy))))
4 df-iso 4796 . 2 (H Isom R, S (A, C) ↔ (H:A1-1-ontoC x A y A (xRy ↔ (Hx)S(Hy))))
52, 3, 43bitr4g 279 1 (B = C → (H Isom R, S (A, B) ↔ H Isom R, S (A, C)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  ∀wral 2614   class class class wbr 4639  –1-1-onto→wf1o 4780   ‘cfv 4781   Isom wiso 4782 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-iso 4796 This theorem is referenced by: (None)
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