Theorem bj-csbprc 37113

Description: More direct proof of csbprc 4362 (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 3851	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		sbcex 3751	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → 𝐴 ∈ V)
3	2	con3i 154	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦 ∈ 𝐵)
4	3	alrimiv 1929	. . 3 ⊢ (¬ 𝐴 ∈ V → ∀𝑦 ¬ [𝐴 / 𝑥]𝑦 ∈ 𝐵)
5		bj-ab0 37111	. . 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 2784	1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2715  Vcvv 3441  [wsbc 3741  ⦋csb 3850  ∅c0 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-nul 4287

This theorem is referenced by: (None)