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Theorem strcollnft 11317
Description: Closed form of strcollnf 11318. Version of ax-strcoll 11315 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 11316 . 2 (∀𝑥𝑎𝑦𝜑 → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑))
2 nfnf1 1479 . . . . 5 𝑏𝑏𝜑
32nfal 1511 . . . 4 𝑏𝑦𝑏𝜑
43nfal 1511 . . 3 𝑏𝑥𝑦𝑏𝜑
5 nfa2 1514 . . . 4 𝑦𝑥𝑦𝑏𝜑
6 nfvd 1465 . . . . 5 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏 𝑦𝑧)
7 nfa1 1477 . . . . . . . 8 𝑥𝑥𝑏𝜑
8 nfcvd 2226 . . . . . . . 8 (∀𝑥𝑏𝜑𝑏𝑎)
9 sp 1444 . . . . . . . 8 (∀𝑥𝑏𝜑 → Ⅎ𝑏𝜑)
107, 8, 9nfrexdxy 2407 . . . . . . 7 (∀𝑥𝑏𝜑 → Ⅎ𝑏𝑥𝑎 𝜑)
1110sps 1473 . . . . . 6 (∀𝑦𝑥𝑏𝜑 → Ⅎ𝑏𝑥𝑎 𝜑)
1211alcoms 1408 . . . . 5 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝑥𝑎 𝜑)
136, 12nfbid 1523 . . . 4 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑))
145, 13nfald 1687 . . 3 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑))
15 nfv 1464 . . . . . 6 𝑦 𝑧 = 𝑏
165, 15nfan 1500 . . . . 5 𝑦(∀𝑥𝑦𝑏𝜑𝑧 = 𝑏)
17 elequ2 1645 . . . . . . 7 (𝑧 = 𝑏 → (𝑦𝑧𝑦𝑏))
1817adantl 271 . . . . . 6 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → (𝑦𝑧𝑦𝑏))
1918bibi1d 231 . . . . 5 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → ((𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ (𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
2016, 19albid 1549 . . . 4 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → (∀𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ ∀𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
2120ex 113 . . 3 (∀𝑥𝑦𝑏𝜑 → (𝑧 = 𝑏 → (∀𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ ∀𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))))
224, 14, 21cbvexd 1847 . 2 (∀𝑥𝑦𝑏𝜑 → (∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
231, 22syl5ib 152 1 (∀𝑥𝑦𝑏𝜑 → (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1285  wnf 1392  wex 1424  wral 2355  wrex 2356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-strcoll 11315
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361
This theorem is referenced by:  strcollnf  11318
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