Theorem nfunid 3899

Description: Deduction version of nfuni 3898. (Contributed by NM, 18-Feb-2013.)

Hypothesis

Ref	Expression
nfunid.3	⊢ (φ → ℲxA)

Assertion

Ref	Expression
nfunid	⊢ (φ → Ⅎx∪A)

Proof of Theorem nfunid

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfuni2 3894	. 2 ⊢ ∪A = {y ∣ ∃z ∈ A y ∈ z}
2		nfv 1619	. . 3 ⊢ Ⅎyφ
3		nfv 1619	. . . 4 ⊢ Ⅎzφ
4		nfunid.3	. . . 4 ⊢ (φ → ℲxA)
5		nfvd 1620	. . . 4 ⊢ (φ → Ⅎx y ∈ z)
6	3, 4, 5	nfrexd 2667	. . 3 ⊢ (φ → Ⅎx∃z ∈ A y ∈ z)
7	2, 6	nfabd 2509	. 2 ⊢ (φ → Ⅎx{y ∣ ∃z ∈ A y ∈ z})
8	1, 7	nfcxfrd 2488	1 ⊢ (φ → Ⅎx∪A)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477  ∃wrex 2616  ∪cuni 3892

This proof depends on 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 proof 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  df-ral 2620  df-rex 2621  df-uni 3893

This theorem is used by:  dfnfc2  3910  nfiotad  4343