Theorem moabexOLD 5427

Description: Obsolete version of moabex 5426 as of 2-Feb-2026. (Contributed by NM, 30-Dec-1996.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
moabexOLD	⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

Proof of Theorem moabexOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfmo 2566	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
2		abss 4010	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑 → 𝑥 ∈ {𝑦}))
3		velsn 4600	. . . . . . 7 ⊢ (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
4	3	imbi2i 339	. . . . . 6 ⊢ ((𝜑 → 𝑥 ∈ {𝑦}) ↔ (𝜑 → 𝑥 = 𝑦))
5	4	albii 1852	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦))
6	2, 5	bitri 278	. . . 4 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦))
7		vsnex 5393	. . . . 5 ⊢ {𝑦} ∈ V
8	7	ssex 5282	. . . 4 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑦} → {𝑥 ∣ 𝜑} ∈ V)
9	6, 8	sylbir 238	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V)
10	9	exlimiv 1963	. 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V)
11	1, 10	sylbi 220	1 ⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812   ∈ wcel 2145  ∃*wmo 2563  {cab 2739  Vcvv 3451   ⊆ wss 3899  {csn 4584

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  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3414  df-v 3453  df-un 3904  df-in 3906  df-ss 3916  df-sn 4585  df-pr 4587

This theorem is used by: (None)