Theorem reu7 2883

Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)

Hypothesis

Ref	Expression
rmo4.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
reu7	⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem reu7

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reu3 2878	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑧)))
2		rmo4.1	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3		equequ1 1689	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
4		equcom 1683	. . . . . . . 8 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦)
5	3, 4	syl6bb 195	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑧 = 𝑦))
6	2, 5	imbi12d 233	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜓 → 𝑧 = 𝑦)))
7	6	cbvralv 2657	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑧) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦))
8	7	rexbii 2445	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑧) ↔ ∃𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦))
9		equequ1 1689	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 ↔ 𝑥 = 𝑦))
10	9	imbi2d 229	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝜓 → 𝑧 = 𝑦) ↔ (𝜓 → 𝑥 = 𝑦)))
11	10	ralbidv 2438	. . . . 5 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
12	11	cbvrexv 2658	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))
13	8, 12	bitri 183	. . 3 ⊢ (∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑧) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))
14	13	anbi2i 453	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑧)) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
15	1, 14	bitri 183	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wral 2417  ∃wrex 2418  ∃!wreu 2419

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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425

This theorem is referenced by: (None)