Theorem ncfineq 4474

Description: Equality theorem for finite cardinality. (Contributed by SF, 20-Jan-2015.)

Assertion

Ref	Expression
ncfineq	⊢ (A = B → Ncfin A = Ncfin B)

Proof of Theorem ncfineq

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2413	. . . 4 ⊢ (A = B → (A ∈ x ↔ B ∈ x))
2	1	anbi2d 684	. . 3 ⊢ (A = B → ((x ∈ Nn ∧ A ∈ x) ↔ (x ∈ Nn ∧ B ∈ x)))
3	2	iotabidv 4361	. 2 ⊢ (A = B → (℩x(x ∈ Nn ∧ A ∈ x)) = (℩x(x ∈ Nn ∧ B ∈ x)))
4		df-ncfin 4443	. 2 ⊢ Ncfin A = (℩x(x ∈ Nn ∧ A ∈ x))
5		df-ncfin 4443	. 2 ⊢ Ncfin B = (℩x(x ∈ Nn ∧ B ∈ x))
6	3, 4, 5	3eqtr4g 2410	1 ⊢ (A = B → Ncfin A = Ncfin B)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ℩cio 4338   Nn cnnc 4374   Nc_fin cncfin 4435

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-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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-uni 3893  df-iota 4340  df-ncfin 4443

This theorem is referenced by:  tncveqnc1fin  4545  vfintle  4547  vfin1cltv  4548  vfinncsp  4555