Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  strcoll2 GIF version

Theorem strcoll2 14018
Description: Version of ax-strcoll 14017 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
Assertion
Ref Expression
strcoll2 (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑))
Distinct variable groups:   𝑎,𝑏,𝑥,𝑦   𝜑,𝑏
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎)

Proof of Theorem strcoll2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 raleq 2665 . . 3 (𝑧 = 𝑎 → (∀𝑥𝑧𝑦𝜑 ↔ ∀𝑥𝑎𝑦𝜑))
2 raleq 2665 . . . . 5 (𝑧 = 𝑎 → (∀𝑥𝑧𝑦𝑏 𝜑 ↔ ∀𝑥𝑎𝑦𝑏 𝜑))
3 rexeq 2666 . . . . . 6 (𝑧 = 𝑎 → (∃𝑥𝑧 𝜑 ↔ ∃𝑥𝑎 𝜑))
43ralbidv 2470 . . . . 5 (𝑧 = 𝑎 → (∀𝑦𝑏𝑥𝑧 𝜑 ↔ ∀𝑦𝑏𝑥𝑎 𝜑))
52, 4anbi12d 470 . . . 4 (𝑧 = 𝑎 → ((∀𝑥𝑧𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑧 𝜑) ↔ (∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑)))
65exbidv 1818 . . 3 (𝑧 = 𝑎 → (∃𝑏(∀𝑥𝑧𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑧 𝜑) ↔ ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑)))
71, 6imbi12d 233 . 2 (𝑧 = 𝑎 → ((∀𝑥𝑧𝑦𝜑 → ∃𝑏(∀𝑥𝑧𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑧 𝜑)) ↔ (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑))))
8 ax-strcoll 14017 . . 3 𝑧(∀𝑥𝑧𝑦𝜑 → ∃𝑏(∀𝑥𝑧𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑧 𝜑))
98spi 1529 . 2 (∀𝑥𝑧𝑦𝜑 → ∃𝑏(∀𝑥𝑧𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑧 𝜑))
107, 9chvarv 1930 1 (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wex 1485  wral 2448  wrex 2449
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-strcoll 14017
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454
This theorem is referenced by:  strcollnft  14019  strcollnfALT  14021
  Copyright terms: Public domain W3C validator