Theorem zfun 7715

Description: Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.) Use ax-un 7714 instead. (New usage is discouraged.)

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 7714	. 2 ⊢ ∃𝑥∀𝑦(∃𝑤(𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥)
2		elequ2 2156	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑥))
3		elequ1 2148	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
4	2, 3	anbi12d 641	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧) ↔ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧)))
5	4	cbvexvw 2056	. . . . 5 ⊢ (∃𝑤(𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧) ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧))
6	5	imbi1i 351	. . . 4 ⊢ ((∃𝑤(𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥) ↔ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
7	6	albii 1838	. . 3 ⊢ (∀𝑦(∃𝑤(𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥) ↔ ∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
8	7	exbii 1867	. 2 ⊢ (∃𝑥∀𝑦(∃𝑤(𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥) ↔ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
9	1, 8	mpbi 232	1 ⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1557  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799

This theorem is referenced by:  axunndlem1  10550