Theorem bj-csbprc 34219

Description: More direct proof of csbprc 4356 (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 3882	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		sbcex 3780	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → 𝐴 ∈ V)
3	2	con3i 157	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦 ∈ 𝐵)
4	3	alrimiv 1922	. . 3 ⊢ (¬ 𝐴 ∈ V → ∀𝑦 ¬ [𝐴 / 𝑥]𝑦 ∈ 𝐵)
5		bj-ab0 34217	. . 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	syl5eq 2866	1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1529   = wceq 1531   ∈ wcel 2108  {cab 2797  Vcvv 3493  [wsbc 3770  ⦋csb 3881  ∅c0 4289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-nul 4290

This theorem is referenced by: (None)