Theorem csbtt 3113

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 3102	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		nfcr 2342	. . . 4 ⊢ (Ⅎ𝑥𝐵 → Ⅎ𝑥 𝑦 ∈ 𝐵)
3		sbctt 3072	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥 𝑦 ∈ 𝐵) → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
4	2, 3	sylan2 286	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
5	4	abbi1dv 2327	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = 𝐵)
6	1, 5	eqtrid 2252	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  Ⅎwnf 1484   ∈ wcel 2178  {cab 2193  Ⅎwnfc 2337  [wsbc 3005  ⦋csb 3101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-sbc 3006  df-csb 3102

This theorem is referenced by:  csbconstgf  3114  sbnfc2  3162