Theorem ifbieq12d 3546

Description: Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
ifbieq12d.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
ifbieq12d.2	⊢ (𝜑 → 𝐴 = 𝐶)
ifbieq12d.3	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
ifbieq12d	⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))

Proof of Theorem ifbieq12d

Step	Hyp	Ref	Expression
1		ifbieq12d.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	ifbid 3541	. 2 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵))
3		ifbieq12d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐶)
4		ifbieq12d.3	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
5	3, 4	ifeq12d 3539	. 2 ⊢ (𝜑 → if(𝜒, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))
6	2, 5	eqtrd 2198	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343  ifcif 3520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-un 3120  df-if 3521

This theorem is referenced by:  updjudhcoinlf  7045  updjudhcoinrg  7046  omp1eom  7060  xaddval  9781  iseqf1olemqval  10422  iseqf1olemqk  10429  seq3f1olemqsum  10435  exp3val  10457  cvgratz  11473  eucalgval2  11985  ennnfonelemg  12336  ennnfonelem1  12340  ressid2  12454  ressval2  12455  lgsval  13545