Theorem sfineq1 4527

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.

Step	Hyp	Ref	Expression
1		eleq1 2413	. . 3 ⊢ (A = B → (A ∈ Nn ↔ B ∈ Nn ))
2		eleq2 2414	. . . . 5 ⊢ (A = B → (℘1y ∈ A ↔ ℘1y ∈ B))
3	2	anbi1d 685	. . . 4 ⊢ (A = B → ((℘1y ∈ A ∧ ℘y ∈ C) ↔ (℘1y ∈ B ∧ ℘y ∈ C)))
4	3	exbidv 1626	. . 3 ⊢ (A = B → (∃y(℘1y ∈ A ∧ ℘y ∈ C) ↔ ∃y(℘1y ∈ B ∧ ℘y ∈ C)))
5	1, 4	3anbi13d 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 4447	. 2 ⊢ ( Sfin (A, C) ↔ (A ∈ Nn ∧ C ∈ Nn ∧ ∃y(℘1y ∈ A ∧ ℘y ∈ C)))
7		df-sfin 4447	. 2 ⊢ ( Sfin (B, C) ↔ (B ∈ Nn ∧ C ∈ Nn ∧ ∃y(℘1y ∈ B ∧ ℘y ∈ C)))
8	5, 6, 7	3bitr4g 279	1 ⊢ (A = B → ( Sfin (A, C) ↔ Sfin (B, C)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ℘cpw 3723  ℘₁cpw1 4136   Nn cnnc 4374   S_fin wsfin 4439

This proof depends on 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 proof depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-cleq 2346  df-clel 2349  df-sfin 4447

This theorem is used by:  sfintfinlem1  4532  sfintfin  4533  spfinsfincl  4540  vfinspsslem1  4551