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Theorem sfineq1 4526
 Description: Equality theorem for the finite S relationship. (Contributed by SF, 27-Jan-2015.)
Assertion
Ref Expression
sfineq1 (A = B → ( Sfin (A, C) ↔ Sfin (B, C)))

Proof of Theorem sfineq1
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eleq1 2413 . . 3 (A = B → (A NnB Nn ))
2 eleq2 2414 . . . . 5 (A = B → (1y A1y B))
32anbi1d 685 . . . 4 (A = B → ((1y A y C) ↔ (1y B y C)))
43exbidv 1626 . . 3 (A = B → (y(1y A y C) ↔ y(1y B y C)))
51, 43anbi13d 1254 . 2 (A = B → ((A Nn C Nn y(1y A y C)) ↔ (B Nn C Nn y(1y B y C))))
6 df-sfin 4446 . 2 ( Sfin (A, C) ↔ (A Nn C Nn y(1y A y C)))
7 df-sfin 4446 . 2 ( Sfin (B, C) ↔ (B Nn C Nn y(1y B y C)))
85, 6, 73bitr4g 279 1 (A = B → ( Sfin (A, C) ↔ Sfin (B, C)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ℘cpw 3722  ℘1cpw1 4135   Nn cnnc 4373   Sfin wsfin 4438 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-cleq 2346  df-clel 2349  df-sfin 4446 This theorem is referenced by:  sfintfinlem1  4531  sfintfin  4532  spfinsfincl  4539  vfinspsslem1  4550
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