Theorem notabw 4313

Description: A class abstraction defined by a negation. Version of notab 4314 using implicit substitution, which does not require ax-10 2141, ax-12 2177. (Contributed by GG, 15-Oct-2024.)

Hypothesis

Ref	Expression
notabw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
notabw	⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑦 ∣ 𝜓})

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem notabw

Step	Hyp	Ref	Expression
1		vex 3484	. . . . 5 ⊢ 𝑥 ∈ V
2	1	biantrur 530	. . . 4 ⊢ (¬ 𝑥 ∈ {𝑦 ∣ 𝜓} ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝜓}))
3		df-clab 2715	. . . . 5 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜓} ↔ [𝑥 / 𝑦]𝜓)
4		notabw.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	4	bicomd 223	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑))
6	5	equcoms 2019	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
7	6	sbievw 2093	. . . . 5 ⊢ ([𝑥 / 𝑦]𝜓 ↔ 𝜑)
8	3, 7	bitri 275	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜓} ↔ 𝜑)
9	2, 8	xchnxbi 332	. . 3 ⊢ (¬ 𝜑 ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝜓}))
10	9	abbii 2809	. 2 ⊢ {𝑥 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝜓})}
11		df-dif 3954	. 2 ⊢ (V ∖ {𝑦 ∣ 𝜓}) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝜓})}
12	10, 11	eqtr4i 2768	1 ⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑦 ∣ 𝜓})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  [wsb 2064   ∈ wcel 2108  {cab 2714  Vcvv 3480   ∖ cdif 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954

This theorem is referenced by:  dfif3  4540