Theorem csbhypf 3172

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2905 for class substitution version. (Contributed by NM, 19-Dec-2008.)

Hypotheses

Ref	Expression
csbhypf.1	⊢ ℲxA
csbhypf.2	⊢ ℲxC
csbhypf.3	⊢ (x = A → B = C)

Assertion

Ref	Expression
csbhypf	⊢ (y = A → [y / x]B = C)

Distinct variable group:   x,y

Allowed substitution hints:   A(x,y)   B(x,y)   C(x,y)

Proof of Theorem csbhypf

Step	Hyp	Ref	Expression
1		csbhypf.1	. . . 4 ⊢ ℲxA
2	1	nfeq2 2501	. . 3 ⊢ Ⅎx y = A
3		nfcsb1v 3169	. . . 4 ⊢ Ⅎx[y / x]B
4		csbhypf.2	. . . 4 ⊢ ℲxC
5	3, 4	nfeq 2497	. . 3 ⊢ Ⅎx[y / x]B = C
6	2, 5	nfim 1813	. 2 ⊢ Ⅎx(y = A → [y / x]B = C)
7		eqeq1 2359	. . 3 ⊢ (x = y → (x = A ↔ y = A))
8		csbeq1a 3145	. . . 4 ⊢ (x = y → B = [y / x]B)
9	8	eqeq1d 2361	. . 3 ⊢ (x = y → (B = C ↔ [y / x]B = C))
10	7, 9	imbi12d 311	. 2 ⊢ (x = y → ((x = A → B = C) ↔ (y = A → [y / x]B = C)))
11		csbhypf.3	. 2 ⊢ (x = A → B = C)
12	6, 10, 11	chvar 1986	1 ⊢ (y = A → [y / x]B = C)

Syntax hints:   → wi 4   = wceq 1642  Ⅎwnfc 2477  [csb 3137

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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-sbc 3048  df-csb 3138

This theorem is referenced by: (None)