Theorem nfiotad 4343

Description: Deduction version of nfiota 4344. (Contributed by NM, 18-Feb-2013.)

Hypotheses

Ref	Expression
nfiotad.1	⊢ Ⅎyφ
nfiotad.2	⊢ (φ → Ⅎxψ)

Assertion

Ref	Expression
nfiotad	⊢ (φ → Ⅎx(℩yψ))

Proof of Theorem nfiotad

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiota2 4341	. 2 ⊢ (℩yψ) = ∪{z ∣ ∀y(ψ ↔ y = z)}
2		nfv 1619	. . . 4 ⊢ Ⅎzφ
3		nfiotad.1	. . . . 5 ⊢ Ⅎyφ
4		nfiotad.2	. . . . . . 7 ⊢ (φ → Ⅎxψ)
5	4	adantr 451	. . . . . 6 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxψ)
6		nfcvf 2512	. . . . . . . 8 ⊢ (¬ ∀x x = y → Ⅎxy)
7	6	adantl 452	. . . . . . 7 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxy)
8		nfcvd 2491	. . . . . . 7 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxz)
9	7, 8	nfeqd 2504	. . . . . 6 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎx y = z)
10	5, 9	nfbid 1832	. . . . 5 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎx(ψ ↔ y = z))
11	3, 10	nfald2 1972	. . . 4 ⊢ (φ → Ⅎx∀y(ψ ↔ y = z))
12	2, 11	nfabd 2509	. . 3 ⊢ (φ → Ⅎx{z ∣ ∀y(ψ ↔ y = z)})
13	12	nfunid 3899	. 2 ⊢ (φ → Ⅎx∪{z ∣ ∀y(ψ ↔ y = z)})
14	1, 13	nfcxfrd 2488	1 ⊢ (φ → Ⅎx(℩yψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642  {cab 2339  Ⅎwnfc 2477  ∪cuni 3892  ℩cio 4338

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-or 359  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  df-ral 2620  df-rex 2621  df-sn 3742  df-uni 3893  df-iota 4340

This theorem is referenced by:  nfiota  4344