Theorem axsep2 4164

Description: A less restrictive version of the Separation Scheme ax-sep 4163, where variables 𝑥 and 𝑧 can both appear free in the wff 𝜑, which can therefore be thought of as 𝜑(𝑥, 𝑧). This version was derived from the more restrictive ax-sep 4163 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

Ref	Expression
axsep2	⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑧)

Proof of Theorem axsep2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2269	. . . . . . 7 ⊢ (𝑤 = 𝑧 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑧))
2	1	anbi1d 465	. . . . . 6 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑧 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑))))
3		anabs5 573	. . . . . 6 ⊢ ((𝑥 ∈ 𝑧 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
4	2, 3	bitrdi 196	. . . . 5 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
5	4	bibi2d 232	. . . 4 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑))) ↔ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
6	5	albidv 1847	. . 3 ⊢ (𝑤 = 𝑧 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑))) ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
7	6	exbidv 1848	. 2 ⊢ (𝑤 = 𝑧 → (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑))) ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
8		ax-sep 4163	. 2 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ (𝑥 ∈ 𝑧 ∧ 𝜑)))
9	7, 8	chvarv 1965	1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wal 1371  ∃wex 1515

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-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-ext 2187  ax-sep 4163

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-cleq 2198  df-clel 2201

This theorem is referenced by: (None)