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Theorem strcollnfALT 11319
Description: Alternate proof of strcollnf 11318, not using strcollnft 11317. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
strcollnf.nf 𝑏𝜑
Assertion
Ref Expression
strcollnfALT (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))
Distinct variable group:   𝑎,𝑏,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem strcollnfALT
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 strcoll2 11316 . 2 (∀𝑥𝑎𝑦𝜑 → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑))
2 nfv 1464 . . . . 5 𝑏 𝑦𝑧
3 nfcv 2225 . . . . . 6 𝑏𝑎
4 strcollnf.nf . . . . . 6 𝑏𝜑
53, 4nfrexxy 2411 . . . . 5 𝑏𝑥𝑎 𝜑
62, 5nfbi 1524 . . . 4 𝑏(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑)
76nfal 1511 . . 3 𝑏𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑)
8 nfv 1464 . . 3 𝑧𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)
9 elequ2 1645 . . . . 5 (𝑧 = 𝑏 → (𝑦𝑧𝑦𝑏))
109bibi1d 231 . . . 4 (𝑧 = 𝑏 → ((𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ (𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
1110albidv 1749 . . 3 (𝑧 = 𝑏 → (∀𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ ∀𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
127, 8, 11cbvex 1683 . 2 (∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))
131, 12sylib 120 1 (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  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: (None)
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