Theorem ifeq2 3667

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

Assertion

Ref	Expression
ifeq2	⊢ (A = B → if(φ, C, A) = if(φ, C, B))

Proof of Theorem ifeq2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabeq 2853	. . 3 ⊢ (A = B → {x ∈ A ∣ ¬ φ} = {x ∈ B ∣ ¬ φ})
2	1	uneq2d 3418	. 2 ⊢ (A = B → ({x ∈ C ∣ φ} ∪ {x ∈ A ∣ ¬ φ}) = ({x ∈ C ∣ φ} ∪ {x ∈ B ∣ ¬ φ}))
3		dfif6 3665	. 2 ⊢ if(φ, C, A) = ({x ∈ C ∣ φ} ∪ {x ∈ A ∣ ¬ φ})
4		dfif6 3665	. 2 ⊢ if(φ, C, B) = ({x ∈ C ∣ φ} ∪ {x ∈ B ∣ ¬ φ})
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → if(φ, C, A) = if(φ, C, B))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642  {crab 2618   ∪ cun 3207   ifcif 3662

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-if 3663

This theorem is referenced by:  ifeq12  3675  ifeq2d  3677  ifbieq2i  3681  ifexg  3721