Theorem mosubt 2957

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 2951	. . . . . 6 ⊢ (𝐴 ∈ V ↔ ∃!𝑦 𝑦 = 𝐴)
2		isset 2783	. . . . . 6 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
3	1, 2	bitr3i 186	. . . . 5 ⊢ (∃!𝑦 𝑦 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
4		nfv 1552	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐴
5	4	euexex 2141	. . . . 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 2131	. . 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 1624	. . . 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 2121	. 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 1371   = wceq 1373  ∃wex 1516  ∃!weu 2055  ∃*wmo 2056   ∈ wcel 2178  Vcvv 2776

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 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-v 2778

This theorem is referenced by:  mosub  2958