Theorem csbtt 3057

Description: Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)

Assertion

Ref	Expression
csbtt	⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)

Proof of Theorem csbtt

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3046	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		nfcr 2300	. . . 4 ⊢ (Ⅎ𝑥𝐵 → Ⅎ𝑥 𝑦 ∈ 𝐵)
3		sbctt 3017	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥 𝑦 ∈ 𝐵) → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
4	2, 3	sylan2 284	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
5	4	abbi1dv 2286	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = 𝐵)
6	1, 5	syl5eq 2211	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  Ⅎwnf 1448   ∈ wcel 2136  {cab 2151  Ⅎwnfc 2295  [wsbc 2951  ⦋csb 3045

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046

This theorem is referenced by:  csbconstgf  3058  sbnfc2  3105