Theorem ifbieq2d 3415

Description: Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypotheses

Ref	Expression
ifbieq2d.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
ifbieq2d.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
ifbieq2d	⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵))

Proof of Theorem ifbieq2d

Step	Hyp	Ref	Expression
1		ifbieq2d.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	ifbid 3412	. 2 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐴))
3		ifbieq2d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
4	3	ifeq2d 3409	. 2 ⊢ (𝜑 → if(𝜒, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵))
5	2, 4	eqtrd 2120	1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵))

Syntax hints:   → wi 4   ↔ wb 103   = wceq 1289  ifcif 3393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-un 3003  df-if 3394

This theorem is referenced by:  xnegeq  9289  iseqf1olemqval  9916  iseqf1olemqk  9923  seq3f1olemqsum  9929  exp3val  9957  gcdval  11229  gcdass  11282  lcmval  11323  lcmass  11345  nnsf  11895  peano4nninf  11896  peano3nninf  11897  exmidsbthr  11913