Theorem csbtt 3148

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 3137	. 2 ⊢ [A / x]B = {y ∣ [̣A / x]̣y ∈ B}
2		nfcr 2481	. . . 4 ⊢ (ℲxB → Ⅎx y ∈ B)
3		sbctt 3108	. . . 4 ⊢ ((A ∈ V ∧ Ⅎx y ∈ B) → ([̣A / x]̣y ∈ B ↔ y ∈ B))
4	2, 3	sylan2 460	. . 3 ⊢ ((A ∈ V ∧ ℲxB) → ([̣A / x]̣y ∈ B ↔ y ∈ B))
5	4	abbi1dv 2469	. 2 ⊢ ((A ∈ V ∧ ℲxB) → {y ∣ [̣A / x]̣y ∈ B} = B)
6	1, 5	syl5eq 2397	1 ⊢ ((A ∈ V ∧ ℲxB) → [A / x]B = B)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476  [̣wsbc 3046  [csb 3136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137

This theorem is referenced by:  csbconstgf  3149  sbnfc2  3196