Theorem ifeq2 4430

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 3431	. . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} = {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})
2	1	uneq2d 4090	. 2 ⊢ (𝐴 = 𝐵 → ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}))
3		dfif6 4428	. 2 ⊢ if(𝜑, 𝐶, 𝐴) = ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})
4		dfif6 4428	. 2 ⊢ if(𝜑, 𝐶, 𝐵) = ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})
5	2, 3, 4	3eqtr4g 2858	1 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538  {crab 3110   ∪ cun 3879  ifcif 4425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-un 3886  df-if 4426

This theorem is referenced by:  ifeq12  4442  ifeq2d  4444  ifbieq2i  4449  somincom  5961  mdetunilem9  21225  prmorcht  25763  pclogsum  25799  noeta  33335  matunitlindflem1  35053  ftc1anclem6  35135  ftc1anclem8  35137  ftc1anc  35138  hdmap1cbv  39098  reabssgn  40336  hoidmv1le  43233  hoidmvlelem3  43236  vonn0ioo  43326  vonn0icc  43327