Theorem ifeq1 4488

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 3417	. . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})
2	1	uneq1d 4126	. 2 ⊢ (𝐴 = 𝐵 → ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑}) = ({𝑥 ∈ 𝐵 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑}))
3		dfif6 4487	. 2 ⊢ if(𝜑, 𝐴, 𝐶) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑})
4		dfif6 4487	. 2 ⊢ if(𝜑, 𝐵, 𝐶) = ({𝑥 ∈ 𝐵 ∣ 𝜑} ∪ {𝑥 ∈ 𝐶 ∣ ¬ 𝜑})
5	2, 3, 4	3eqtr4g 2789	1 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540  {crab 3402   ∪ cun 3909  ifcif 4484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-un 3916  df-if 4485

This theorem is referenced by:  ifeq12  4503  ifeq1d  4504  ifbieq12i  4512  rdgeq2  8357  dfoi  9440  wemaplem2  9476  cantnflem1  9618  prodeq2w  15852  prodeq2ii  15853  mgm2nsgrplem2  18822  mgm2nsgrplem3  18823  mplcoe3  21921  marrepval0  22424  ellimc  25750  ply1nzb  26004  dchrvmasumiflem1  27388  signspval  34516  dfrdg2  35756  sumeq2si  36163  prodeq2si  36165  dfafv22  47233