Theorem bj-csbprc 36876

Description: More direct proof of csbprc 4432 (fewer essential steps). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-csbprc	⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)

Proof of Theorem bj-csbprc

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3922	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		sbcex 3814	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → 𝐴 ∈ V)
3	2	con3i 154	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦 ∈ 𝐵)
4	3	alrimiv 1926	. . 3 ⊢ (¬ 𝐴 ∈ V → ∀𝑦 ¬ [𝐴 / 𝑥]𝑦 ∈ 𝐵)
5		bj-ab0 36874	. . 3 ⊢ (∀𝑦 ¬ [𝐴 / 𝑥]𝑦 ∈ 𝐵 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = ∅)
6	4, 5	syl 17	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = ∅)
7	1, 6	eqtrid 2792	1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535   = wceq 1537   ∈ wcel 2108  {cab 2717  Vcvv 3488  [wsbc 3804  ⦋csb 3921  ∅c0 4352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-nul 4353

This theorem is referenced by: (None)