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Theorem sfineq2 4527
 Description: Equality theorem for the finite S relationship. (Contributed by SF, 27-Jan-2015.)
Assertion
Ref Expression
sfineq2 (A = B → ( Sfin (C, A) ↔ Sfin (C, B)))

Proof of Theorem sfineq2
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 → (y Ay B))
32anbi2d 684 . . . 4 (A = B → ((1y C y A) ↔ (1y C y B)))
43exbidv 1626 . . 3 (A = B → (y(1y C y A) ↔ y(1y C y B)))
51, 43anbi23d 1255 . 2 (A = B → ((C Nn A Nn y(1y C y A)) ↔ (C Nn B Nn y(1y C y B))))
6 df-sfin 4446 . 2 ( Sfin (C, A) ↔ (C Nn A Nn y(1y C y A)))
7 df-sfin 4446 . 2 ( Sfin (C, B) ↔ (C Nn B Nn y(1y C y B)))
85, 6, 73bitr4g 279 1 (A = B → ( Sfin (C, A) ↔ Sfin (C, B)))
 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-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-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  t1csfin1c  4545  vfinspss  4551
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