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Theorem strcollnft 9033
Description: Closed form of strcollnf 9034. Version of ax-strcoll 9031 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.)
Assertion
Ref Expression
strcollnft (xy𝑏φ → (x 𝑎 yφ𝑏y(y 𝑏x 𝑎 φ)))
Distinct variable group:   𝑎,𝑏,x,y
Allowed substitution hints:   φ(x,y,𝑎,𝑏)

Proof of Theorem strcollnft
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 strcoll2 9032 . 2 (x 𝑎 yφzy(y zx 𝑎 φ))
2 nfnf1 1433 . . . . 5 𝑏𝑏φ
32nfal 1465 . . . 4 𝑏y𝑏φ
43nfal 1465 . . 3 𝑏xy𝑏φ
5 nfa2 1468 . . . 4 yxy𝑏φ
6 nfvd 1419 . . . . 5 (xy𝑏φ → Ⅎ𝑏 y z)
7 nfa1 1431 . . . . . . . 8 xx𝑏φ
8 nfcvd 2176 . . . . . . . 8 (x𝑏φ𝑏𝑎)
9 sp 1398 . . . . . . . 8 (x𝑏φ → Ⅎ𝑏φ)
107, 8, 9nfrexdxy 2351 . . . . . . 7 (x𝑏φ → Ⅎ𝑏x 𝑎 φ)
1110sps 1427 . . . . . 6 (yx𝑏φ → Ⅎ𝑏x 𝑎 φ)
1211alcoms 1362 . . . . 5 (xy𝑏φ → Ⅎ𝑏x 𝑎 φ)
136, 12nfbid 1477 . . . 4 (xy𝑏φ → Ⅎ𝑏(y zx 𝑎 φ))
145, 13nfald 1640 . . 3 (xy𝑏φ → Ⅎ𝑏y(y zx 𝑎 φ))
15 nfv 1418 . . . . . 6 y z = 𝑏
165, 15nfan 1454 . . . . 5 y(xy𝑏φ z = 𝑏)
17 elequ2 1598 . . . . . . 7 (z = 𝑏 → (y zy 𝑏))
1817adantl 262 . . . . . 6 ((xy𝑏φ z = 𝑏) → (y zy 𝑏))
1918bibi1d 222 . . . . 5 ((xy𝑏φ z = 𝑏) → ((y zx 𝑎 φ) ↔ (y 𝑏x 𝑎 φ)))
2016, 19albid 1503 . . . 4 ((xy𝑏φ z = 𝑏) → (y(y zx 𝑎 φ) ↔ y(y 𝑏x 𝑎 φ)))
2120ex 108 . . 3 (xy𝑏φ → (z = 𝑏 → (y(y zx 𝑎 φ) ↔ y(y 𝑏x 𝑎 φ))))
224, 14, 21cbvexd 1799 . 2 (xy𝑏φ → (zy(y zx 𝑎 φ) ↔ 𝑏y(y 𝑏x 𝑎 φ)))
231, 22syl5ib 143 1 (xy𝑏φ → (x 𝑎 yφ𝑏y(y 𝑏x 𝑎 φ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wnf 1346  wex 1378  wral 2300  wrex 2301
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-strcoll 9031
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306
This theorem is referenced by:  strcollnf  9034
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