Theorem notabw 4204

Description: A class abstraction defined by a negation. Version of notab 4205 using implicit substitution, which does not require ax-10 2143, ax-12 2177. (Contributed by Gino Giotto, 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 3402	. . . . 5 ⊢ 𝑥 ∈ V
2	1	biantrur 534	. . . 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 226	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑))
6	5	equcoms 2030	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
7	6	sbievw 2101	. . . . 5 ⊢ ([𝑥 / 𝑦]𝜓 ↔ 𝜑)
8	3, 7	bitri 278	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜓} ↔ 𝜑)
9	2, 8	xchnxbi 335	. . 3 ⊢ (¬ 𝜑 ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝜓}))
10	9	abbii 2801	. 2 ⊢ {𝑥 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝜓})}
11		df-dif 3856	. 2 ⊢ (V ∖ {𝑦 ∣ 𝜓}) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝜓})}
12	10, 11	eqtr4i 2762	1 ⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑦 ∣ 𝜓})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543  [wsb 2072   ∈ wcel 2112  {cab 2714  Vcvv 3398   ∖ cdif 3850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-v 3400  df-dif 3856

This theorem is referenced by:  dfif3  4439