Theorem nfif 3687

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	⊢ Ⅎx if(φ, A, B)

Proof of Theorem nfif

Step	Hyp	Ref	Expression
1		nfif.1	. . . 4 ⊢ Ⅎxφ
2	1	a1i 10	. . 3 ⊢ ( ⊤ → Ⅎxφ)
3		nfif.2	. . . 4 ⊢ ℲxA
4	3	a1i 10	. . 3 ⊢ ( ⊤ → ℲxA)
5		nfif.3	. . . 4 ⊢ ℲxB
6	5	a1i 10	. . 3 ⊢ ( ⊤ → ℲxB)
7	2, 4, 6	nfifd 3686	. 2 ⊢ ( ⊤ → Ⅎx if(φ, A, B))
8	7	trud 1323	1 ⊢ Ⅎx if(φ, A, B)

This proof depends on syntax axioms:   ⊤ wtru 1316  Ⅎwnf 1544  Ⅎwnfc 2477   ifcif 3663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-if 3664

This theorem is used by:  csbifg  3691