Theorem cnvkeq 4216

Description: Equality theorem for Kuratowski converse. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
cnvkeq	⊢ (A = B → ◡kA = ◡kB)

Proof of Theorem cnvkeq

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . . 5 ⊢ (A = B → (⟪z, y⟫ ∈ A ↔ ⟪z, y⟫ ∈ B))
2	1	anbi2d 684	. . . 4 ⊢ (A = B → ((x = ⟪y, z⟫ ∧ ⟪z, y⟫ ∈ A) ↔ (x = ⟪y, z⟫ ∧ ⟪z, y⟫ ∈ B)))
3	2	2exbidv 1628	. . 3 ⊢ (A = B → (∃y∃z(x = ⟪y, z⟫ ∧ ⟪z, y⟫ ∈ A) ↔ ∃y∃z(x = ⟪y, z⟫ ∧ ⟪z, y⟫ ∈ B)))
4	3	abbidv 2468	. 2 ⊢ (A = B → {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ ⟪z, y⟫ ∈ A)} = {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ ⟪z, y⟫ ∈ B)})
5		df-cnvk 4187	. 2 ⊢ ◡kA = {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ ⟪z, y⟫ ∈ A)}
6		df-cnvk 4187	. 2 ⊢ ◡kB = {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ ⟪z, y⟫ ∈ B)}
7	4, 5, 6	3eqtr4g 2410	1 ⊢ (A = B → ◡kA = ◡kB)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ⟪copk 4058  ^◡_kccnvk 4176

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  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-cnvk 4187

This theorem is used by:  cnvkeqi  4217  cnvkeqd  4218  cokeq2  4232  cnvkexg  4287