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Theorem isoeq4 5485
 Description: Equality theorem for isomorphisms. (Contributed by set.mm contributors, 17-May-2004.)
Assertion
Ref Expression
isoeq4 (A = C → (H Isom R, S (A, B) ↔ H Isom R, S (C, B)))

Proof of Theorem isoeq4
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oeq2 5282 . . 3 (A = C → (H:A1-1-ontoBH:C1-1-ontoB))
2 raleq 2807 . . . 4 (A = C → (y A (xRy ↔ (Hx)S(Hy)) ↔ y C (xRy ↔ (Hx)S(Hy))))
32raleqbi1dv 2815 . . 3 (A = C → (x A y A (xRy ↔ (Hx)S(Hy)) ↔ x C y C (xRy ↔ (Hx)S(Hy))))
41, 3anbi12d 691 . 2 (A = C → ((H:A1-1-ontoB x A y A (xRy ↔ (Hx)S(Hy))) ↔ (H:C1-1-ontoB x C y C (xRy ↔ (Hx)S(Hy)))))
5 df-iso 4796 . 2 (H Isom R, S (A, B) ↔ (H:A1-1-ontoB x A y A (xRy ↔ (Hx)S(Hy))))
6 df-iso 4796 . 2 (H Isom R, S (C, B) ↔ (H:C1-1-ontoB x C y C (xRy ↔ (Hx)S(Hy))))
74, 5, 63bitr4g 279 1 (A = C → (H Isom R, S (A, B) ↔ H Isom R, S (C, B)))
 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-fn 4790  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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