Theorem csbhypf 3856

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3500 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 2972	. . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴
3		nfcsb1v 3852	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
4		csbhypf.2	. . . 4 ⊢ Ⅎ𝑥𝐶
5	3, 4	nfeq 2968	. . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 = 𝐶
6	2, 5	nfim 1897	. 2 ⊢ Ⅎ𝑥(𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)
7		eqeq1 2802	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
8		csbeq1a 3842	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
9	8	eqeq1d 2800	. . 3 ⊢ (𝑥 = 𝑦 → (𝐵 = 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 = 𝐶))
10	7, 9	imbi12d 348	. 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	chvarfv 2240	1 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   = wceq 1538  Ⅎwnfc 2936  ⦋csb 3828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-sbc 3721  df-csb 3829

This theorem is referenced by:  disji2  5012  disjprgw  5025  disjprg  5026  disjxun  5028  tfisi  7553  coe1fzgsumdlem  20930  evl1gsumdlem  20980  iundisj2  24153  disji2f  30340  disjif2  30344  iundisj2f  30353  iundisj2fi  30546