Theorem notabw 4259

Description: A class abstraction defined by a negation. Version of notab 4260 using implicit substitution, which does not require ax-10 2178, ax-12 2213. (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 3454	. . . . 5 ⊢ 𝑥 ∈ V
2	1	biantrur 540	. . . 4 ⊢ (¬ 𝑥 ∈ {𝑦 ∣ 𝜓} ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝜓}))
3		df-clab 2739	. . . . 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 2053	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
7	6	sbievw 2131	. . . . 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 2827	. 2 ⊢ {𝑥 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝜓})}
11		df-dif 3902	. 2 ⊢ (V ∖ {𝑦 ∣ 𝜓}) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝜓})}
12	10, 11	eqtr4i 2786	1 ⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑦 ∣ 𝜓})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  [wsb 2099   ∈ wcel 2145  {cab 2738  Vcvv 3450   ∖ cdif 3896

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 3902

This theorem is used by:  dfif3  4497