Theorem eusvobj1 7405

Description: Specify the same object in two ways when class 𝐵(𝑦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Hypothesis

Ref	Expression
eusvobj1.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
eusvobj1	⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (℩𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) = (℩𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem eusvobj1

Step	Hyp	Ref	Expression
1		nfeu1 2615	. . 3 ⊢ Ⅎ𝑥∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵
2		eusvobj1.1	. . . 4 ⊢ 𝐵 ∈ V
3	2	eusvobj2 7404	. . 3 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
4	1, 3	alrimi 2250	. 2 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
5		iotabi 6500	. 2 ⊢ (∀𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → (℩𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) = (℩𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
6	4, 5	syl 18	1 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (℩𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) = (℩𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃!weu 2594  ∀wral 3077  ∃wrex 3087  Vcvv 3451  ℩cio 6485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-ss 3916  df-nul 4280  df-sn 4585  df-uni 4868  df-iota 6487

This theorem is used by: (None)