Theorem bm1.3ii 5226

Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 5223. 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 1957	. . 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 841	. . . . 5 ⊢ (((𝜑 → 𝑦 ∈ 𝑧) ∧ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → (𝑦 ∈ 𝑥 ↔ 𝜑))
3	2	alanimi 1819	. . . 4 ⊢ ((∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
4	3	eximi 1837	. . 3 ⊢ (∃𝑥(∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
5	1, 4	sylbir 234	. 2 ⊢ ((∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
6		bm1.3ii.1	. . . 4 ⊢ ∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥)
7		elequ2 2121	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
8	7	imbi2d 341	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝜑 → 𝑦 ∈ 𝑥) ↔ (𝜑 → 𝑦 ∈ 𝑧)))
9	8	albidv 1923	. . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝜑 → 𝑦 ∈ 𝑥) ↔ ∀𝑦(𝜑 → 𝑦 ∈ 𝑧)))
10	9	cbvexvw 2040	. . . 4 ⊢ (∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥) ↔ ∃𝑧∀𝑦(𝜑 → 𝑦 ∈ 𝑧))
11	6, 10	mpbi 229	. . 3 ⊢ ∃𝑧∀𝑦(𝜑 → 𝑦 ∈ 𝑧)
12		ax-sep 5223	. . 3 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))
13	11, 12	exan 1865	. 2 ⊢ ∃𝑧(∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑)))
14	5, 13	exlimiiv 1934	1 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-sep 5223

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783

This theorem is referenced by:  axpow3  5291  vpwex  5300  axprlem4  5349  zfpair2  5353  axun2  7590  uniex2  7591