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Theorem exss 5320
 Description: Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)
Assertion
Ref Expression
exss (∃𝑥𝐴 𝜑 → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem exss
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-rab 3115 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
21neeq1i 3051 . . 3 ({𝑥𝐴𝜑} ≠ ∅ ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ≠ ∅)
3 rabn0 4293 . . 3 ({𝑥𝐴𝜑} ≠ ∅ ↔ ∃𝑥𝐴 𝜑)
4 n0 4260 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ≠ ∅ ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
52, 3, 43bitr3i 304 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
6 vex 3444 . . . . . 6 𝑧 ∈ V
76snss 4679 . . . . 5 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ {𝑧} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)})
8 ssab2 4006 . . . . . 6 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
9 sstr2 3922 . . . . . 6 ({𝑧} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)} → ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴 → {𝑧} ⊆ 𝐴))
108, 9mpi 20 . . . . 5 ({𝑧} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)} → {𝑧} ⊆ 𝐴)
117, 10sylbi 220 . . . 4 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → {𝑧} ⊆ 𝐴)
12 simpr 488 . . . . . . . 8 (([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) → [𝑧 / 𝑥]𝜑)
13 equsb1v 2109 . . . . . . . . 9 [𝑧 / 𝑥]𝑥 = 𝑧
14 velsn 4541 . . . . . . . . . 10 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
1514sbbii 2081 . . . . . . . . 9 ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ↔ [𝑧 / 𝑥]𝑥 = 𝑧)
1613, 15mpbir 234 . . . . . . . 8 [𝑧 / 𝑥]𝑥 ∈ {𝑧}
1712, 16jctil 523 . . . . . . 7 (([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) → ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
18 df-clab 2777 . . . . . . . 8 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ [𝑧 / 𝑥](𝑥𝐴𝜑))
19 sban 2085 . . . . . . . 8 ([𝑧 / 𝑥](𝑥𝐴𝜑) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
2018, 19bitri 278 . . . . . . 7 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
21 df-rab 3115 . . . . . . . . 9 {𝑥 ∈ {𝑧} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)}
2221eleq2i 2881 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)})
23 df-clab 2777 . . . . . . . 8 (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑))
24 sban 2085 . . . . . . . 8 ([𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
2522, 23, 243bitri 300 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
2617, 20, 253imtr4i 295 . . . . . 6 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → 𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑})
2726ne0d 4251 . . . . 5 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → {𝑥 ∈ {𝑧} ∣ 𝜑} ≠ ∅)
28 rabn0 4293 . . . . 5 ({𝑥 ∈ {𝑧} ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ {𝑧}𝜑)
2927, 28sylib 221 . . . 4 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∃𝑥 ∈ {𝑧}𝜑)
30 snex 5297 . . . . 5 {𝑧} ∈ V
31 sseq1 3940 . . . . . 6 (𝑦 = {𝑧} → (𝑦𝐴 ↔ {𝑧} ⊆ 𝐴))
32 rexeq 3359 . . . . . 6 (𝑦 = {𝑧} → (∃𝑥𝑦 𝜑 ↔ ∃𝑥 ∈ {𝑧}𝜑))
3331, 32anbi12d 633 . . . . 5 (𝑦 = {𝑧} → ((𝑦𝐴 ∧ ∃𝑥𝑦 𝜑) ↔ ({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑)))
3430, 33spcev 3555 . . . 4 (({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑) → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
3511, 29, 34syl2anc 587 . . 3 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
3635exlimiv 1931 . 2 (∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
375, 36sylbi 220 1 (∃𝑥𝐴 𝜑 → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781  [wsb 2069   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∃wrex 3107  {crab 3110   ⊆ wss 3881  ∅c0 4243  {csn 4525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528 This theorem is referenced by: (None)
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