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Theorem sscoll2 11050
Description: Version of ax-sscoll 11049 with two DV conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)
Assertion
Ref Expression
sscoll2 𝑐𝑧(∀𝑥𝑎𝑦𝑏 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑎 𝜑))
Distinct variable groups:   𝑎,𝑏,𝑐,𝑑,𝑥,𝑦,𝑧   𝜑,𝑐,𝑑
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑎,𝑏)

Proof of Theorem sscoll2
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1462 . . 3 𝑐(𝑢 = 𝑎𝑣 = 𝑏)
2 nfv 1462 . . . 4 𝑧(𝑢 = 𝑎𝑣 = 𝑏)
3 simpl 107 . . . . . 6 ((𝑢 = 𝑎𝑣 = 𝑏) → 𝑢 = 𝑎)
4 rexeq 2555 . . . . . . 7 (𝑣 = 𝑏 → (∃𝑦𝑣 𝜑 ↔ ∃𝑦𝑏 𝜑))
54adantl 271 . . . . . 6 ((𝑢 = 𝑎𝑣 = 𝑏) → (∃𝑦𝑣 𝜑 ↔ ∃𝑦𝑏 𝜑))
63, 5raleqbidv 2566 . . . . 5 ((𝑢 = 𝑎𝑣 = 𝑏) → (∀𝑥𝑢𝑦𝑣 𝜑 ↔ ∀𝑥𝑎𝑦𝑏 𝜑))
7 nfv 1462 . . . . . 6 𝑑(𝑢 = 𝑎𝑣 = 𝑏)
8 nfv 1462 . . . . . . 7 𝑦(𝑢 = 𝑎𝑣 = 𝑏)
9 rexeq 2555 . . . . . . . . 9 (𝑢 = 𝑎 → (∃𝑥𝑢 𝜑 ↔ ∃𝑥𝑎 𝜑))
109adantr 270 . . . . . . . 8 ((𝑢 = 𝑎𝑣 = 𝑏) → (∃𝑥𝑢 𝜑 ↔ ∃𝑥𝑎 𝜑))
1110bibi2d 230 . . . . . . 7 ((𝑢 = 𝑎𝑣 = 𝑏) → ((𝑦𝑑 ↔ ∃𝑥𝑢 𝜑) ↔ (𝑦𝑑 ↔ ∃𝑥𝑎 𝜑)))
128, 11albid 1547 . . . . . 6 ((𝑢 = 𝑎𝑣 = 𝑏) → (∀𝑦(𝑦𝑑 ↔ ∃𝑥𝑢 𝜑) ↔ ∀𝑦(𝑦𝑑 ↔ ∃𝑥𝑎 𝜑)))
137, 12rexbid 2372 . . . . 5 ((𝑢 = 𝑎𝑣 = 𝑏) → (∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑢 𝜑) ↔ ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑎 𝜑)))
146, 13imbi12d 232 . . . 4 ((𝑢 = 𝑎𝑣 = 𝑏) → ((∀𝑥𝑢𝑦𝑣 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑢 𝜑)) ↔ (∀𝑥𝑎𝑦𝑏 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑎 𝜑))))
152, 14albid 1547 . . 3 ((𝑢 = 𝑎𝑣 = 𝑏) → (∀𝑧(∀𝑥𝑢𝑦𝑣 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑢 𝜑)) ↔ ∀𝑧(∀𝑥𝑎𝑦𝑏 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑎 𝜑))))
161, 15exbid 1548 . 2 ((𝑢 = 𝑎𝑣 = 𝑏) → (∃𝑐𝑧(∀𝑥𝑢𝑦𝑣 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑢 𝜑)) ↔ ∃𝑐𝑧(∀𝑥𝑎𝑦𝑏 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑎 𝜑))))
17 ax-sscoll 11049 . . . 4 𝑢𝑣𝑐𝑧(∀𝑥𝑢𝑦𝑣 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑢 𝜑))
1817spi 1470 . . 3 𝑣𝑐𝑧(∀𝑥𝑢𝑦𝑣 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑢 𝜑))
1918spi 1470 . 2 𝑐𝑧(∀𝑥𝑢𝑦𝑣 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑢 𝜑))
2016, 19ch2varv 10839 1 𝑐𝑧(∀𝑥𝑎𝑦𝑏 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑎 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283  wex 1422  wral 2353  wrex 2354
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sscoll 11049
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359
This theorem is referenced by: (None)
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