Theorem mosubt 2907

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 2901	. . . . . 6 ⊢ (𝐴 ∈ V ↔ ∃!𝑦 𝑦 = 𝐴)
2		isset 2736	. . . . . 6 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
3	1, 2	bitr3i 185	. . . . 5 ⊢ (∃!𝑦 𝑦 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
4		nfv 1521	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐴
5	4	euexex 2104	. . . . 5 ⊢ ((∃!𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
6	3, 5	sylanbr 283	. . . 4 ⊢ ((∃𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
7	6	expcom 115	. . 3 ⊢ (∀𝑦∃*𝑥𝜑 → (∃𝑦 𝑦 = 𝐴 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)))
8		moanimv 2094	. . 3 ⊢ (∃*𝑥(∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) ↔ (∃𝑦 𝑦 = 𝐴 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)))
9	7, 8	sylibr 133	. 2 ⊢ (∀𝑦∃*𝑥𝜑 → ∃*𝑥(∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑)))
10		simpl 108	. . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝜑) → 𝑦 = 𝐴)
11	10	eximi 1593	. . . 4 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) → ∃𝑦 𝑦 = 𝐴)
12	11	ancri 322	. . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) → (∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑)))
13	12	moimi 2084	. 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 103  ∀wal 1346   = wceq 1348  ∃wex 1485  ∃!weu 2019  ∃*wmo 2020   ∈ wcel 2141  Vcvv 2730

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  mosub  2908