Theorem csbhypf 3878

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3512 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 2941	. . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴
3		nfcsb1v 3874	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
4		csbhypf.2	. . . 4 ⊢ Ⅎ𝑥𝐶
5	3, 4	nfeq 2937	. . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 = 𝐶
6	2, 5	nfim 1929	. 2 ⊢ Ⅎ𝑥(𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)
7		eqeq1 2766	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
8		csbeq1a 3864	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
9	8	eqeq1d 2764	. . 3 ⊢ (𝑥 = 𝑦 → (𝐵 = 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 = 𝐶))
10	7, 9	imbi12d 347	. 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 2278	1 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Ⅎwnfc 2909  ⦋csb 3850

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-sbc 3743  df-csb 3851

This theorem is used by:  disji2  5091  disjprg  5103  disjxun  5105  tfisi  7858  coe1fzgsumdlem  22532  evl1gsumdlem  22585  iundisj2  25781  disji2f  33052  disjif2  33056  iundisj2f  33065  iundisj2fi  33270  evl1gprodd  42985