Theorem ifeq1 4460

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 3408	. . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})
2	1	uneq1d 4092	. 2 ⊢ (𝐴 = 𝐵 → ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑}) = ({𝑥 ∈ 𝐵 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑}))
3		dfif6 4459	. 2 ⊢ if(𝜑, 𝐴, 𝐶) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑})
4		dfif6 4459	. 2 ⊢ if(𝜑, 𝐵, 𝐶) = ({𝑥 ∈ 𝐵 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑})
5	2, 3, 4	3eqtr4g 2804	1 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539  {crab 3067   ∪ cun 3881  ifcif 4456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-un 3888  df-if 4457

This theorem is referenced by:  ifeq12  4474  ifeq1d  4475  ifbieq12i  4483  rdgeq2  8214  dfoi  9200  wemaplem2  9236  cantnflem1  9377  prodeq2w  15550  prodeq2ii  15551  mgm2nsgrplem2  18473  mgm2nsgrplem3  18474  mplcoe3  21149  marrepval0  21618  ellimc  24942  ply1nzb  25192  dchrvmasumiflem1  26554  signspval  32431  dfrdg2  33677  dfafv22  44638