Theorem ifeq1 1778

Description: Equality theorem for conditional operator.

Assertion

Ref	Expression
ifeq1	⊢ (A = B → if(φ, A, C) = if(φ, B, C))

Proof of Theorem ifeq1

Step	Hyp	Ref	Expression
1		eleq2 1150	. . . . 5 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
2	1	anbi1d 469	. . . 4 ⊢ (A = B → ((x ∈ A ∧ φ) ↔ (x ∈ B ∧ φ)))
3	2	orbi1d 467	. . 3 ⊢ (A = B → (((x ∈ A ∧ φ) ∨ (x ∈ C ∧ ¬ φ)) ↔ ((x ∈ B ∧ φ) ∨ (x ∈ C ∧ ¬ φ))))
4	3	biabdv 1183	. 2 ⊢ (A = B → {x∣((x ∈ A ∧ φ) ∨ (x ∈ C ∧ ¬ φ))} = {x∣((x ∈ B ∧ φ) ∨ (x ∈ C ∧ ¬ φ))})
5		df-if 1777	. 2 ⊢ if(φ, A, C) = {x∣((x ∈ A ∧ φ) ∨ (x ∈ C ∧ ¬ φ))}
6		df-if 1777	. 2 ⊢ if(φ, B, C) = {x∣((x ∈ B ∧ φ) ∨ (x ∈ C ∧ ¬ φ))}
7	4, 5, 6	3eqtr4g 1147	1 ⊢ (A = B → if(φ, A, C) = if(φ, B, C))

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  ifcif 1776

This theorem is referenced by:  ifeq12 1782  oev 3122  ruclem4 4888

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-if 1777

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