Theorem nfiundg 49178

Description: Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2375, see nfiund 49177 for a weaker version that does not require it. (Contributed by Emmett Weisz, 6-Dec-2019.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfiundg.1	⊢ Ⅎ𝑥𝜑
nfiundg.2	⊢ (𝜑 → Ⅎ𝑦𝐴)
nfiundg.3	⊢ (𝜑 → Ⅎ𝑦𝐵)

Assertion

Ref	Expression
nfiundg	⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵)

Proof of Theorem nfiundg

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 4973	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfv 1913	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfiundg.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4		nfiundg.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑦𝐴)
5		nfiundg.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝐵)
6	5	nfcrd 2891	. . . 4 ⊢ (𝜑 → Ⅎ𝑦 𝑧 ∈ 𝐵)
7	3, 4, 6	nfrexd 3356	. . 3 ⊢ (𝜑 → Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
8	2, 7	nfabd 2920	. 2 ⊢ (𝜑 → Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵})
9	1, 8	nfcxfrd 2896	1 ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4  Ⅎwnf 1782   ∈ wcel 2107  {cab 2712  Ⅎwnfc 2882  ∃wrex 3059  ∪ ciun 4971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2375  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-iun 4973

This theorem is referenced by: (None)