Theorem csbhypf 3585

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3284 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 2809	. . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴
3		nfcsb1v 3582	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
4		csbhypf.2	. . . 4 ⊢ Ⅎ𝑥𝐶
5	3, 4	nfeq 2805	. . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 = 𝐶
6	2, 5	nfim 1865	. 2 ⊢ Ⅎ𝑥(𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)
7		eqeq1 2655	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
8		csbeq1a 3575	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
9	8	eqeq1d 2653	. . 3 ⊢ (𝑥 = 𝑦 → (𝐵 = 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 = 𝐶))
10	7, 9	imbi12d 333	. 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 2298	1 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   = wceq 1523  Ⅎwnfc 2780  ⦋csb 3566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-sbc 3469  df-csb 3567

This theorem is referenced by:  disji2  4668  disjprg  4680  disjxun  4683  tfisi  7100  coe1fzgsumdlem  19719  evl1gsumdlem  19768  iundisj2  23363  disji2f  29516  disjif2  29520  iundisj2f  29529  iundisj2fi  29684