Theorem mosubt 3003

Description: "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.)

Assertion

Ref	Expression
mosubt	⊢ (∀𝑦∃*𝑥𝜑 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mosubt

Step	Hyp	Ref	Expression
1		eueq 2997	. . . . . 6 ⊢ (𝐴 ∈ V ↔ ∃!𝑦 𝑦 = 𝐴)
2		isset 2828	. . . . . 6 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
3	1, 2	bitr3i 186	. . . . 5 ⊢ (∃!𝑦 𝑦 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
4		nfv 1581	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐴
5	4	euexex 2172	. . . . 5 ⊢ ((∃!𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
6	3, 5	sylanbr 285	. . . 4 ⊢ ((∃𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
7	6	expcom 116	. . 3 ⊢ (∀𝑦∃*𝑥𝜑 → (∃𝑦 𝑦 = 𝐴 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)))
8		moanimv 2162	. . 3 ⊢ (∃*𝑥(∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) ↔ (∃𝑦 𝑦 = 𝐴 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)))
9	7, 8	sylibr 134	. 2 ⊢ (∀𝑦∃*𝑥𝜑 → ∃*𝑥(∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑)))
10		simpl 109	. . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝜑) → 𝑦 = 𝐴)
11	10	eximi 1653	. . . 4 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) → ∃𝑦 𝑦 = 𝐴)
12	11	ancri 324	. . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) → (∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑)))
13	12	moimi 2152	. 2 ⊢ (∃*𝑥(∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
14	9, 13	syl 14	1 ⊢ (∀𝑦∃*𝑥𝜑 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1400   = wceq 1402  ∃wex 1545  ∃!weu 2086  ∃*wmo 2087   ∈ wcel 2209  Vcvv 2821

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823

This theorem is referenced by:  mosub  3004