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Theorem strcollnf 10110
 Description: Version of ax-strcoll 10107 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 (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))
Distinct variable group:   𝑎,𝑏,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem strcollnf
StepHypRef Expression
1 strcollnft 10109 . 2 (∀𝑥𝑦𝑏𝜑 → (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
2 strcollnf.nf . . 3 𝑏𝜑
32ax-gen 1338 . 2 𝑦𝑏𝜑
41, 3mpg 1340 1 (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381  ∀wral 2306  ∃wrex 2307 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-strcoll 10107 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312 This theorem is referenced by: (None)
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