Theorem zfun 4171

Description: Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)

Assertion

Ref	Expression
zfun	⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem zfun

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-un 4170	. 2 ⊢ ∃𝑥∀𝑦(∃𝑤(𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥)
2		elequ2 1601	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑥))
3		elequ1 1600	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
4	2, 3	anbi12d 442	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧) ↔ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧)))
5	4	cbvexv 1795	. . . . 5 ⊢ (∃𝑤(𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧) ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
6	5	imbi1i 227	. . . 4 ⊢ ((∃𝑤(𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥) ↔ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
7	6	albii 1359	. . 3 ⊢ (∀𝑦(∃𝑤(𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥) ↔ ∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
8	7	exbii 1496	. 2 ⊢ (∃𝑥∀𝑦(∃𝑤(𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥) ↔ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
9	1, 8	mpbi 133	1 ⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-un 4170

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  uniex2  4173  bj-uniex2  10036