Theorem csbhypf 3087

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2779 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 2324	. . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴
3		nfcsb1v 3082	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
4		csbhypf.2	. . . 4 ⊢ Ⅎ𝑥𝐶
5	3, 4	nfeq 2320	. . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 = 𝐶
6	2, 5	nfim 1565	. 2 ⊢ Ⅎ𝑥(𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)
7		eqeq1 2177	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
8		csbeq1a 3058	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
9	8	eqeq1d 2179	. . 3 ⊢ (𝑥 = 𝑦 → (𝐵 = 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 = 𝐶))
10	7, 9	imbi12d 233	. 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 1750	1 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   = wceq 1348  Ⅎwnfc 2299  ⦋csb 3049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-sbc 2956  df-csb 3050

This theorem is referenced by:  disji2  3982  tfisi  4571