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Theorem strcollnf 9415
 Description: Version of ax-strcoll 9412 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.)
Hypothesis
Ref Expression
strcollnf.nf 𝑏φ
Assertion
Ref Expression
strcollnf (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 strcollnf
StepHypRef Expression
1 strcollnft 9414 . 2 (xy𝑏φ → (x 𝑎 yφ𝑏y(y 𝑏x 𝑎 φ)))
2 strcollnf.nf . . 3 𝑏φ
32ax-gen 1335 . 2 y𝑏φ
41, 3mpg 1337 1 (x 𝑎 yφ𝑏y(y 𝑏x 𝑎 φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240  Ⅎwnf 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 9412 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: (None)
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