Theorem ifeq1 4487

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 3431	. . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})
2	1	uneq1d 4123	. 2 ⊢ (𝐴 = 𝐵 → ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑}) = ({𝑥 ∈ 𝐵 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑}))
3		dfif6 4486	. 2 ⊢ if(𝜑, 𝐴, 𝐶) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑})
4		dfif6 4486	. 2 ⊢ if(𝜑, 𝐵, 𝐶) = ({𝑥 ∈ 𝐵 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑})
5	2, 3, 4	3eqtr4g 2825	1 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1563  {crab 3417   ∪ cun 3905  ifcif 4483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-un 3912  df-if 4484

This theorem is referenced by:  ifeq12  4502  ifeq1d  4503  ifbieq12i  4511  rdgeq2  8387  dfoi  9461  wemaplem2  9497  cantnflem1  9646  prodeq2w  15952  prodeq2ii  15953  mgm2nsgrplem2  18969  mgm2nsgrplem3  18970  mplcoe3  22146  marrepval0  22675  ellimc  25989  ply1nzb  26237  dchrvmasumiflem1  27619  signspval  34851  dfrdg2  36151  sumeq2si  36570  prodeq2si  36572  dfafv22  47852