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Theorem exss 3933
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 (x A φy(yA x y 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 3218 . . 3 (z z {x Aφ} ↔ x A φ)
2 df-rab 2289 . . . . 5 {x Aφ} = {x ∣ (x A φ)}
32eleq2i 2082 . . . 4 (z {x Aφ} ↔ z {x ∣ (x A φ)})
43exbii 1474 . . 3 (z z {x Aφ} ↔ z z {x ∣ (x A φ)})
51, 4bitr3i 175 . 2 (x A φz z {x ∣ (x A φ)})
6 vex 2534 . . . . . 6 z V
76snss 3464 . . . . 5 (z {x ∣ (x A φ)} ↔ {z} ⊆ {x ∣ (x A φ)})
8 ssab2 2997 . . . . . 6 {x ∣ (x A φ)} ⊆ A
9 sstr2 2925 . . . . . 6 ({z} ⊆ {x ∣ (x A φ)} → ({x ∣ (x A φ)} ⊆ A → {z} ⊆ A))
108, 9mpi 15 . . . . 5 ({z} ⊆ {x ∣ (x A φ)} → {z} ⊆ A)
117, 10sylbi 114 . . . 4 (z {x ∣ (x A φ)} → {z} ⊆ A)
12 simpr 103 . . . . . . . 8 (([z / x]x A [z / x]φ) → [z / x]φ)
13 equsb1 1646 . . . . . . . . 9 [z / x]x = z
14 elsn 3361 . . . . . . . . . 10 (x {z} ↔ x = z)
1514sbbii 1626 . . . . . . . . 9 ([z / x]x {z} ↔ [z / x]x = z)
1613, 15mpbir 134 . . . . . . . 8 [z / x]x {z}
1712, 16jctil 295 . . . . . . 7 (([z / x]x A [z / x]φ) → ([z / x]x {z} [z / x]φ))
18 df-clab 2005 . . . . . . . 8 (z {x ∣ (x A φ)} ↔ [z / x](x A φ))
19 sban 1807 . . . . . . . 8 ([z / x](x A φ) ↔ ([z / x]x A [z / x]φ))
2018, 19bitri 173 . . . . . . 7 (z {x ∣ (x A φ)} ↔ ([z / x]x A [z / x]φ))
21 df-rab 2289 . . . . . . . . 9 {x {z} ∣ φ} = {x ∣ (x {z} φ)}
2221eleq2i 2082 . . . . . . . 8 (z {x {z} ∣ φ} ↔ z {x ∣ (x {z} φ)})
23 df-clab 2005 . . . . . . . . 9 (z {x ∣ (x {z} φ)} ↔ [z / x](x {z} φ))
24 sban 1807 . . . . . . . . 9 ([z / x](x {z} φ) ↔ ([z / x]x {z} [z / x]φ))
2523, 24bitri 173 . . . . . . . 8 (z {x ∣ (x {z} φ)} ↔ ([z / x]x {z} [z / x]φ))
2622, 25bitri 173 . . . . . . 7 (z {x {z} ∣ φ} ↔ ([z / x]x {z} [z / x]φ))
2717, 20, 263imtr4i 190 . . . . . 6 (z {x ∣ (x A φ)} → z {x {z} ∣ φ})
28 elex2 2543 . . . . . 6 (z {x {z} ∣ φ} → w w {x {z} ∣ φ})
2927, 28syl 14 . . . . 5 (z {x ∣ (x A φ)} → w w {x {z} ∣ φ})
30 rabn0m 3218 . . . . 5 (w w {x {z} ∣ φ} ↔ x {z}φ)
3129, 30sylib 127 . . . 4 (z {x ∣ (x A φ)} → x {z}φ)
32 snexgOLD 3905 . . . . . 6 (z V → {z} V)
336, 32ax-mp 7 . . . . 5 {z} V
34 sseq1 2939 . . . . . 6 (y = {z} → (yA ↔ {z} ⊆ A))
35 rexeq 2480 . . . . . 6 (y = {z} → (x y φx {z}φ))
3634, 35anbi12d 445 . . . . 5 (y = {z} → ((yA x y φ) ↔ ({z} ⊆ A x {z}φ)))
3733, 36spcev 2620 . . . 4 (({z} ⊆ A x {z}φ) → y(yA x y φ))
3811, 31, 37syl2anc 393 . . 3 (z {x ∣ (x A φ)} → y(yA x y φ))
3938exlimiv 1467 . 2 (z z {x ∣ (x A φ)} → y(yA x y φ))
405, 39sylbi 114 1 (x A φy(yA x y φ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1226  wex 1358   wcel 1370  [wsb 1623  {cab 2004  wrex 2281  {crab 2284  Vcvv 2531  wss 2890  {csn 3346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-rab 2289  df-v 2533  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352
This theorem is referenced by: (None)
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