Theorem eusvobj1 5938

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 2066	. . 3 ⊢ Ⅎ𝑥∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵
2		eusvobj1.1	. . . 4 ⊢ 𝐵 ∈ V
3	2	eusvobj2 5937	. . 3 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
4	1, 3	alrimi 1546	. 2 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
5		iotabi 5246	. 2 ⊢ (∀𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → (℩𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) = (℩𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
6	4, 5	syl 14	1 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (℩𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) = (℩𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1371   = wceq 1373  ∃!weu 2055   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  Vcvv 2773  ℩cio 5235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3000  df-csb 3095  df-sn 3640  df-uni 3853  df-iota 5237

This theorem is referenced by: (None)