Theorem bm1.3ii 4103

Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4100. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-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		bm1.3ii.1	. . . . 5 ⊢ ∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥)
2		elequ2 2141	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
3	2	imbi2d 229	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝜑 → 𝑦 ∈ 𝑥) ↔ (𝜑 → 𝑦 ∈ 𝑧)))
4	3	albidv 1812	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝜑 → 𝑦 ∈ 𝑥) ↔ ∀𝑦(𝜑 → 𝑦 ∈ 𝑧)))
5	4	cbvexv 1906	. . . . 5 ⊢ (∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥) ↔ ∃𝑧∀𝑦(𝜑 → 𝑦 ∈ 𝑧))
6	1, 5	mpbi 144	. . . 4 ⊢ ∃𝑧∀𝑦(𝜑 → 𝑦 ∈ 𝑧)
7		ax-sep 4100	. . . 4 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))
8	6, 7	pm3.2i 270	. . 3 ⊢ (∃𝑧∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑)))
9	8	exan 1681	. 2 ⊢ ∃𝑧(∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑)))
10		19.42v 1894	. . . 4 ⊢ (∃𝑥(∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) ↔ (∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))))
11		bimsc1 953	. . . . . 6 ⊢ (((𝜑 → 𝑦 ∈ 𝑧) ∧ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → (𝑦 ∈ 𝑥 ↔ 𝜑))
12	11	alanimi 1447	. . . . 5 ⊢ ((∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
13	12	eximi 1588	. . . 4 ⊢ (∃𝑥(∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
14	10, 13	sylbir 134	. . 3 ⊢ ((∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
15	14	exlimiv 1586	. 2 ⊢ (∃𝑧(∀𝑦(𝜑 → 𝑦 ∈ 𝑧) ∧ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ 𝜑))) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
16	9, 15	ax-mp 5	1 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-14 2139  ax-sep 4100

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  axpow3  4156  vpwex  4158  zfpair2  4188  axun2  4413  uniex2  4414