Theorem ifeq2 4490

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 3430	. . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} = {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})
2	1	uneq2d 4170	. 2 ⊢ (𝐴 = 𝐵 → ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}))
3		dfif6 4488	. 2 ⊢ if(𝜑, 𝐶, 𝐴) = ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})
4		dfif6 4488	. 2 ⊢ if(𝜑, 𝐶, 𝐵) = ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})
5	2, 3, 4	3eqtr4g 2823	1 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570  {crab 3416   ∪ cun 4152  ifcif 4485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-un 4154  df-if 4486

This theorem is used by:  ifeq12  4504  ifeq2d  4506  ifbieq2i  4511  somincom  6146  mdetunilem9  22990  matunitlindflem1  23049  prmorcht  27562  pclogsum  27599  ftc1anclem6  38884  ftc1anclem8  38886  ftc1anc  38887  hdmap1cbv  43125  reabssgn  44897  hoidmv1le  47877  hoidmvlelem3  47880  vonn0ioo  47970  vonn0icc  47971