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Theorem strcollnft 10496
Description: Closed form of strcollnf 10497. Version of ax-strcoll 10494 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 → (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 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 10495 . 2 (∀𝑥𝑎𝑦𝜑 → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑))
2 nfnf1 1452 . . . . 5 𝑏𝑏𝜑
32nfal 1484 . . . 4 𝑏𝑦𝑏𝜑
43nfal 1484 . . 3 𝑏𝑥𝑦𝑏𝜑
5 nfa2 1487 . . . 4 𝑦𝑥𝑦𝑏𝜑
6 nfvd 1438 . . . . 5 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏 𝑦𝑧)
7 nfa1 1450 . . . . . . . 8 𝑥𝑥𝑏𝜑
8 nfcvd 2195 . . . . . . . 8 (∀𝑥𝑏𝜑𝑏𝑎)
9 sp 1417 . . . . . . . 8 (∀𝑥𝑏𝜑 → Ⅎ𝑏𝜑)
107, 8, 9nfrexdxy 2374 . . . . . . 7 (∀𝑥𝑏𝜑 → Ⅎ𝑏𝑥𝑎 𝜑)
1110sps 1446 . . . . . 6 (∀𝑦𝑥𝑏𝜑 → Ⅎ𝑏𝑥𝑎 𝜑)
1211alcoms 1381 . . . . 5 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝑥𝑎 𝜑)
136, 12nfbid 1496 . . . 4 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑))
145, 13nfald 1659 . . 3 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑))
15 nfv 1437 . . . . . 6 𝑦 𝑧 = 𝑏
165, 15nfan 1473 . . . . 5 𝑦(∀𝑥𝑦𝑏𝜑𝑧 = 𝑏)
17 elequ2 1617 . . . . . . 7 (𝑧 = 𝑏 → (𝑦𝑧𝑦𝑏))
1817adantl 266 . . . . . 6 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → (𝑦𝑧𝑦𝑏))
1918bibi1d 226 . . . . 5 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → ((𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ (𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
2016, 19albid 1522 . . . 4 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → (∀𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ ∀𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
2120ex 112 . . 3 (∀𝑥𝑦𝑏𝜑 → (𝑧 = 𝑏 → (∀𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ ∀𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))))
224, 14, 21cbvexd 1818 . 2 (∀𝑥𝑦𝑏𝜑 → (∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
231, 22syl5ib 147 1 (∀𝑥𝑦𝑏𝜑 → (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  wnf 1365  wex 1397  wral 2323  wrex 2324
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-strcoll 10494
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329
This theorem is referenced by:  strcollnf  10497
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