Theorem csbifg 3691

Description: Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
csbifg	⊢ (A ∈ V → [A / x] if(φ, B, C) = if([̣A / x]̣φ, [A / x]B, [A / x]C))

Proof of Theorem csbifg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3140	. . 3 ⊢ (y = A → [y / x] if(φ, B, C) = [A / x] if(φ, B, C))
2		dfsbcq2 3050	. . . 4 ⊢ (y = A → ([y / x]φ ↔ [̣A / x]̣φ))
3		csbeq1 3140	. . . 4 ⊢ (y = A → [y / x]B = [A / x]B)
4		csbeq1 3140	. . . 4 ⊢ (y = A → [y / x]C = [A / x]C)
5	2, 3, 4	ifbieq12d 3685	. . 3 ⊢ (y = A → if([y / x]φ, [y / x]B, [y / x]C) = if([̣A / x]̣φ, [A / x]B, [A / x]C))
6	1, 5	eqeq12d 2367	. 2 ⊢ (y = A → ([y / x] if(φ, B, C) = if([y / x]φ, [y / x]B, [y / x]C) ↔ [A / x] if(φ, B, C) = if([̣A / x]̣φ, [A / x]B, [A / x]C)))
7		vex 2863	. . 3 ⊢ y ∈ V
8		nfs1v 2106	. . . 4 ⊢ Ⅎx[y / x]φ
9		nfcsb1v 3169	. . . 4 ⊢ Ⅎx[y / x]B
10		nfcsb1v 3169	. . . 4 ⊢ Ⅎx[y / x]C
11	8, 9, 10	nfif 3687	. . 3 ⊢ Ⅎx if([y / x]φ, [y / x]B, [y / x]C)
12		sbequ12 1919	. . . 4 ⊢ (x = y → (φ ↔ [y / x]φ))
13		csbeq1a 3145	. . . 4 ⊢ (x = y → B = [y / x]B)
14		csbeq1a 3145	. . . 4 ⊢ (x = y → C = [y / x]C)
15	12, 13, 14	ifbieq12d 3685	. . 3 ⊢ (x = y → if(φ, B, C) = if([y / x]φ, [y / x]B, [y / x]C))
16	7, 11, 15	csbief 3178	. 2 ⊢ [y / x] if(φ, B, C) = if([y / x]φ, [y / x]B, [y / x]C)
17	6, 16	vtoclg 2915	1 ⊢ (A ∈ V → [A / x] if(φ, B, C) = if([̣A / x]̣φ, [A / x]B, [A / x]C))

Syntax hints:   → wi 4   = wceq 1642  [wsb 1648   ∈ wcel 1710  [̣wsbc 3047  [csb 3137   ifcif 3663

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  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-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-un 3215  df-if 3664

This theorem is referenced by: (None)