Theorem nfabd2 2508

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)

Hypotheses

Ref	Expression
nfabd2.1	⊢ Ⅎyφ
nfabd2.2	⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxψ)

Assertion

Ref	Expression
nfabd2	⊢ (φ → Ⅎx{y ∣ ψ})

Proof of Theorem nfabd2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . 4 ⊢ Ⅎz(φ ∧ ¬ ∀x x = y)
2		df-clab 2340	. . . . 5 ⊢ (z ∈ {y ∣ ψ} ↔ [z / y]ψ)
3		nfabd2.1	. . . . . . 7 ⊢ Ⅎyφ
4		nfnae 1956	. . . . . . 7 ⊢ Ⅎy ¬ ∀x x = y
5	3, 4	nfan 1824	. . . . . 6 ⊢ Ⅎy(φ ∧ ¬ ∀x x = y)
6		nfabd2.2	. . . . . 6 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxψ)
7	5, 6	nfsbd 2111	. . . . 5 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎx[z / y]ψ)
8	2, 7	nfxfrd 1571	. . . 4 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎx z ∈ {y ∣ ψ})
9	1, 8	nfcd 2485	. . 3 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎx{y ∣ ψ})
10	9	ex 423	. 2 ⊢ (φ → (¬ ∀x x = y → Ⅎx{y ∣ ψ}))
11		nfab1 2492	. . 3 ⊢ Ⅎy{y ∣ ψ}
12		eqidd 2354	. . . 4 ⊢ (∀x x = y → {y ∣ ψ} = {y ∣ ψ})
13	12	drnfc1 2506	. . 3 ⊢ (∀x x = y → (Ⅎx{y ∣ ψ} ↔ Ⅎy{y ∣ ψ}))
14	11, 13	mpbiri 224	. 2 ⊢ (∀x x = y → Ⅎx{y ∣ ψ})
15	10, 14	pm2.61d2 152	1 ⊢ (φ → Ⅎx{y ∣ ψ})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479

This theorem is referenced by:  nfabd  2509  nfrab  2793