Theorem ifeq1 4458

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 3405	. . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})
2	1	uneq1d 4097	. 2 ⊢ (𝐴 = 𝐵 → ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑}) = ({𝑥 ∈ 𝐵 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑}))
3		dfif6 4457	. 2 ⊢ if(𝜑, 𝐴, 𝐶) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑})
4		dfif6 4457	. 2 ⊢ if(𝜑, 𝐵, 𝐶) = ({𝑥 ∈ 𝐵 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑})
5	2, 3, 4	3eqtr4g 2799	1 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1547  {crab 3391   ∪ cun 3881  ifcif 4454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-un 3888  df-if 4455

This theorem is referenced by:  ifeq12  4473  ifeq1d  4474  ifbieq12i  4482  rdgeq2  8341  dfoi  9416  wemaplem2  9452  cantnflem1  9601  prodeq2w  15866  prodeq2ii  15867  mgm2nsgrplem2  18881  mgm2nsgrplem3  18882  mplcoe3  22014  marrepval0  22544  ellimc  25858  ply1nzb  26106  dchrvmasumiflem1  27482  signspval  34736  dfrdg2  36021  sumeq2si  36430  prodeq2si  36432  dfafv22  47722