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Theorem strcollnft 13016
Description: Closed form of strcollnf 13017. Version of ax-strcoll 13014 with one disjoint variable condition removed, the other disjoint variable condition replaced with 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 13015 . 2 (∀𝑥𝑎𝑦𝜑 → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑))
2 nfnf1 1506 . . . . 5 𝑏𝑏𝜑
32nfal 1538 . . . 4 𝑏𝑦𝑏𝜑
43nfal 1538 . . 3 𝑏𝑥𝑦𝑏𝜑
5 nfa2 1541 . . . 4 𝑦𝑥𝑦𝑏𝜑
6 nfvd 1492 . . . . 5 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏 𝑦𝑧)
7 nfa1 1504 . . . . . . . 8 𝑥𝑥𝑏𝜑
8 nfcvd 2257 . . . . . . . 8 (∀𝑥𝑏𝜑𝑏𝑎)
9 sp 1471 . . . . . . . 8 (∀𝑥𝑏𝜑 → Ⅎ𝑏𝜑)
107, 8, 9nfrexdxy 2443 . . . . . . 7 (∀𝑥𝑏𝜑 → Ⅎ𝑏𝑥𝑎 𝜑)
1110sps 1500 . . . . . 6 (∀𝑦𝑥𝑏𝜑 → Ⅎ𝑏𝑥𝑎 𝜑)
1211alcoms 1435 . . . . 5 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝑥𝑎 𝜑)
136, 12nfbid 1550 . . . 4 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑))
145, 13nfald 1716 . . 3 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑))
15 nfv 1491 . . . . . 6 𝑦 𝑧 = 𝑏
165, 15nfan 1527 . . . . 5 𝑦(∀𝑥𝑦𝑏𝜑𝑧 = 𝑏)
17 elequ2 1674 . . . . . . 7 (𝑧 = 𝑏 → (𝑦𝑧𝑦𝑏))
1817adantl 273 . . . . . 6 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → (𝑦𝑧𝑦𝑏))
1918bibi1d 232 . . . . 5 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → ((𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ (𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
2016, 19albid 1577 . . . 4 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → (∀𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ ∀𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
2120ex 114 . . 3 (∀𝑥𝑦𝑏𝜑 → (𝑧 = 𝑏 → (∀𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ ∀𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))))
224, 14, 21cbvexd 1877 . 2 (∀𝑥𝑦𝑏𝜑 → (∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
231, 22syl5ib 153 1 (∀𝑥𝑦𝑏𝜑 → (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312  wnf 1419  wex 1451  wral 2391  wrex 2392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-strcoll 13014
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397
This theorem is referenced by:  strcollnf  13017
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