Theorem bj-axsep2 32921

Description: Remove dependency on ax-12 2047 and ax-13 2246 from axsep2 4782 while shortening its proof. Remark: the comment in zfauscl 4783 is misleading: the essential use of ax-ext 2602 is the one via eleq2 2690 and not the one via vtocl 3259, since the latter can be proved without ax-ext 2602 (see bj-vtocl 32909). (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-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 bj-axsep2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ2 2004	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑧))
2	1	anbi1d 741	. . . . 5 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑤 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
3	2	bibi2d 332	. . . 4 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
4	3	albidv 1849	. . 3 ⊢ (𝑤 = 𝑧 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
5	4	exbidv 1850	. 2 ⊢ (𝑤 = 𝑧 → (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
6		ax-sep 4781	. 2 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑))
7	5, 6	bj-chvarvv 32726	1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Syntax hints:   ↔ wb 196   ∧ wa 384  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)