Theorem notabw 4248

Description: A class abstraction defined by a negation. Version of notab 4249 using implicit substitution, which does not require ax-10 2152, ax-12 2189. (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 3436	. . . . 5 ⊢ 𝑥 ∈ V
2	1	biantrur 535	. . . 4 ⊢ (¬ 𝑥 ∈ {𝑦 ∣ 𝜓} ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝜓}))
3		df-clab 2719	. . . . 5 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜓} ↔ [𝑥 / 𝑦]𝜓)
4		notabw.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	4	bicomd 224	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑))
6	5	equcoms 2027	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
7	6	sbievw 2104	. . . . 5 ⊢ ([𝑥 / 𝑦]𝜓 ↔ 𝜑)
8	3, 7	bitri 276	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜓} ↔ 𝜑)
9	2, 8	xchnxbi 333	. . 3 ⊢ (¬ 𝜑 ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝜓}))
10	9	abbii 2807	. 2 ⊢ {𝑥 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝜓})}
11		df-dif 3893	. 2 ⊢ (V ∖ {𝑦 ∣ 𝜓}) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝜓})}
12	10, 11	eqtr4i 2766	1 ⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑦 ∣ 𝜓})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  [wsb 2073   ∈ wcel 2119  {cab 2718  Vcvv 3432   ∖ cdif 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-dif 3893

This theorem is referenced by:  dfif3  4476