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Theorem strcollnfALT 12995
Description: Alternate proof of strcollnf 12994, not using strcollnft 12993. (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 12992 . 2 (∀𝑥𝑎𝑦𝜑 → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑))
2 nfv 1491 . . . . 5 𝑏 𝑦𝑧
3 nfcv 2256 . . . . . 6 𝑏𝑎
4 strcollnf.nf . . . . . 6 𝑏𝜑
53, 4nfrexxy 2447 . . . . 5 𝑏𝑥𝑎 𝜑
62, 5nfbi 1551 . . . 4 𝑏(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑)
76nfal 1538 . . 3 𝑏𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑)
8 nfv 1491 . . 3 𝑧𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)
9 elequ2 1674 . . . . 5 (𝑧 = 𝑏 → (𝑦𝑧𝑦𝑏))
109bibi1d 232 . . . 4 (𝑧 = 𝑏 → ((𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ (𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
1110albidv 1778 . . 3 (𝑧 = 𝑏 → (∀𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ ∀𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
127, 8, 11cbvex 1712 . 2 (∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥𝑎 𝜑) ↔ ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))
131, 12sylib 121 1 (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  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 12991
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: (None)
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