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Theorem exss 3991
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 variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabn0m 3273 . . 3 (∃𝑧 𝑧 ∈ {𝑥𝐴𝜑} ↔ ∃𝑥𝐴 𝜑)
2 df-rab 2332 . . . . 5 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
32eleq2i 2120 . . . 4 (𝑧 ∈ {𝑥𝐴𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
43exbii 1512 . . 3 (∃𝑧 𝑧 ∈ {𝑥𝐴𝜑} ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
51, 4bitr3i 179 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
6 vex 2577 . . . . . 6 𝑧 ∈ V
76snss 3522 . . . . 5 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ {𝑧} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)})
8 ssab2 3052 . . . . . 6 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
9 sstr2 2980 . . . . . 6 ({𝑧} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)} → ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴 → {𝑧} ⊆ 𝐴))
108, 9mpi 15 . . . . 5 ({𝑧} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)} → {𝑧} ⊆ 𝐴)
117, 10sylbi 118 . . . 4 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → {𝑧} ⊆ 𝐴)
12 simpr 107 . . . . . . . 8 (([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) → [𝑧 / 𝑥]𝜑)
13 equsb1 1684 . . . . . . . . 9 [𝑧 / 𝑥]𝑥 = 𝑧
14 velsn 3420 . . . . . . . . . 10 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
1514sbbii 1664 . . . . . . . . 9 ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ↔ [𝑧 / 𝑥]𝑥 = 𝑧)
1613, 15mpbir 138 . . . . . . . 8 [𝑧 / 𝑥]𝑥 ∈ {𝑧}
1712, 16jctil 299 . . . . . . 7 (([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) → ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
18 df-clab 2043 . . . . . . . 8 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ [𝑧 / 𝑥](𝑥𝐴𝜑))
19 sban 1845 . . . . . . . 8 ([𝑧 / 𝑥](𝑥𝐴𝜑) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
2018, 19bitri 177 . . . . . . 7 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
21 df-rab 2332 . . . . . . . . 9 {𝑥 ∈ {𝑧} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)}
2221eleq2i 2120 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)})
23 df-clab 2043 . . . . . . . . 9 (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑))
24 sban 1845 . . . . . . . . 9 ([𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
2523, 24bitri 177 . . . . . . . 8 (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
2622, 25bitri 177 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
2717, 20, 263imtr4i 194 . . . . . 6 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → 𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑})
28 elex2 2587 . . . . . 6 (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} → ∃𝑤 𝑤 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑})
2927, 28syl 14 . . . . 5 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∃𝑤 𝑤 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑})
30 rabn0m 3273 . . . . 5 (∃𝑤 𝑤 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ ∃𝑥 ∈ {𝑧}𝜑)
3129, 30sylib 131 . . . 4 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∃𝑥 ∈ {𝑧}𝜑)
32 snexgOLD 3963 . . . . . 6 (𝑧 ∈ V → {𝑧} ∈ V)
336, 32ax-mp 7 . . . . 5 {𝑧} ∈ V
34 sseq1 2994 . . . . . 6 (𝑦 = {𝑧} → (𝑦𝐴 ↔ {𝑧} ⊆ 𝐴))
35 rexeq 2523 . . . . . 6 (𝑦 = {𝑧} → (∃𝑥𝑦 𝜑 ↔ ∃𝑥 ∈ {𝑧}𝜑))
3634, 35anbi12d 450 . . . . 5 (𝑦 = {𝑧} → ((𝑦𝐴 ∧ ∃𝑥𝑦 𝜑) ↔ ({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑)))
3733, 36spcev 2664 . . . 4 (({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑) → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
3811, 31, 37syl2anc 397 . . 3 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
3938exlimiv 1505 . 2 (∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
405, 39sylbi 118 1 (∃𝑥𝐴 𝜑 → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397  wcel 1409  [wsb 1661  {cab 2042  wrex 2324  {crab 2327  Vcvv 2574  wss 2945  {csn 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-rab 2332  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409
This theorem is referenced by: (None)
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