Theorem nfiunya 3944

Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)

Hypotheses

Ref	Expression
nfiunya.1	⊢ Ⅎ𝑦𝐴
nfiunya.2	⊢ Ⅎ𝑦𝐵

Assertion

Ref	Expression
nfiunya	⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfiunya

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 3918	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfiunya.1	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfiunya.2	. . . . 5 ⊢ Ⅎ𝑦𝐵
4	3	nfcri 2333	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5	2, 4	nfrexya 2538	. . 3 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfab 2344	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
7	1, 6	nfcxfr 2336	1 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2167  {cab 2182  Ⅎwnfc 2326  ∃wrex 2476  ∪ ciun 3916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-iun 3918

This theorem is referenced by: (None)