Theorem bm1.3ii 5010

Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 5007. Similar to Theorem 1.3(ii) of [BellMachover] p. 463. (Contributed by NM, 21-Jun-1993.)

Hypothesis

Ref	Expression
bm1.3ii.1	⊢ ∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥)

Assertion

Ref	Expression
bm1.3ii	⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)

Distinct variable groups:   𝜑,𝑥   𝑥,𝑦

Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem bm1.3ii

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.42v 2052	. . 3 ⊢ (∃𝑥(∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) ↔ (∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))))
2		bimsc1 875	. . . . 5 ⊢ (((𝜑 → 𝑦 ∈ 𝑧) ∧ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → (𝑦 ∈ 𝑥 ↔ 𝜑))
3	2	alanimi 1915	. . . 4 ⊢ ((∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
4	3	eximi 1933	. . 3 ⊢ (∃𝑥(∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
5	1, 4	sylbir 227	. 2 ⊢ ((∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
6		bm1.3ii.1	. . . . 5 ⊢ ∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥)
7		elequ2 2178	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
8	7	imbi2d 332	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝜑 → 𝑦 ∈ 𝑥) ↔ (𝜑 → 𝑦 ∈ 𝑧)))
9	8	albidv 2019	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝜑 → 𝑦 ∈ 𝑥) ↔ ∀𝑦(𝜑 → 𝑦 ∈ 𝑧)))
10	9	cbvexvw 2144	. . . . 5 ⊢ (∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥) ↔ ∃𝑧∀𝑦(𝜑 → 𝑦 ∈ 𝑧))
11	6, 10	mpbi 222	. . . 4 ⊢ ∃𝑧∀𝑦(𝜑 → 𝑦 ∈ 𝑧)
12		ax-sep 5007	. . . 4 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))
13	11, 12	pm3.2i 464	. . 3 ⊢ (∃𝑧∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑)))
14	13	exan 1961	. 2 ⊢ ∃𝑧(∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑)))
15	5, 14	exlimiiv 2030	1 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1654  ∃wex 1878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-sep 5007

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879

This theorem is referenced by:  axpow3  5070  vpwex  5079  zfpair2  5130  axun2  7216  uniex2  7217