Theorem nfif 3650

Description: Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Hypotheses

Ref	Expression
nfif.1	⊢ Ⅎ𝑥𝜑
nfif.2	⊢ Ⅎ𝑥𝐴
nfif.3	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfif	⊢ Ⅎ𝑥if(𝜑, 𝐴, 𝐵)

Proof of Theorem nfif

Step	Hyp	Ref	Expression
1		nfif.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
3		nfif.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4	3	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
5		nfif.3	. . . 4 ⊢ Ⅎ𝑥𝐵
6	5	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐵)
7	2, 4, 6	nfifd 3649	. 2 ⊢ (⊤ → Ⅎ𝑥if(𝜑, 𝐴, 𝐵))
8	7	mptru 1407	1 ⊢ Ⅎ𝑥if(𝜑, 𝐴, 𝐵)

Syntax hints:  ⊤wtru 1399  Ⅎwnf 1509  Ⅎwnfc 2371  ifcif 3619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-if 3620

This theorem is referenced by:  nfsum1  12037  nfsum  12038  sumrbdclem  12059  summodclem2a  12063  zsumdc  12066  fsum3  12069  isumss  12073  isumss2  12075  fsum3cvg2  12076  nfcprod1  12236  nfcprod  12237  cbvprod  12240  prodrbdclem  12253  prodmodclem2a  12258  zproddc  12261  fprodseq  12265  fprodntrivap  12266  prodssdc  12271  pcmpt  13037  pcmptdvds  13039