Theorem ifeq1 3667

Description: Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

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

Proof of Theorem ifeq1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabeq 2854	. . 3 ⊢ (A = B → {x ∈ A ∣ φ} = {x ∈ B ∣ φ})
2	1	uneq1d 3418	. 2 ⊢ (A = B → ({x ∈ A ∣ φ} ∪ {x ∈ C ∣ ¬ φ}) = ({x ∈ B ∣ φ} ∪ {x ∈ C ∣ ¬ φ}))
3		dfif6 3666	. 2 ⊢ if(φ, A, C) = ({x ∈ A ∣ φ} ∪ {x ∈ C ∣ ¬ φ})
4		dfif6 3666	. 2 ⊢ if(φ, B, C) = ({x ∈ B ∣ φ} ∪ {x ∈ C ∣ ¬ φ})
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → if(φ, A, C) = if(φ, B, C))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642  {crab 2619   ∪ cun 3208   ifcif 3663

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-nan 1288  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-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-if 3664

This theorem is referenced by:  ifeq12  3676  ifeq1d  3677  ifbieq12i  3684  ifexg  3722