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Theorem sn-axprlem3 39350
Description: axprlem3 5303 using only Tarski's FOL axiom schemes and ax-rep 5166. (Contributed by SN, 22-Sep-2023.)
Assertion
Ref Expression
sn-axprlem3 𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏))
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝜑,𝑦,𝑧   𝑦,𝑎,𝑧   𝑦,𝑏,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑤,𝑎,𝑏)

Proof of Theorem sn-axprlem3
StepHypRef Expression
1 axrep6 5173 . 2 (∀𝑤∃*𝑧if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏)))
2 ax6evr 2022 . . . . 5 𝑦 𝑎 = 𝑦
3 ifptru 1071 . . . . . . . . . 10 (𝜑 → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) ↔ 𝑧 = 𝑎))
43biimpd 232 . . . . . . . . 9 (𝜑 → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑎))
5 equtrr 2029 . . . . . . . . 9 (𝑎 = 𝑦 → (𝑧 = 𝑎𝑧 = 𝑦))
64, 5sylan9r 512 . . . . . . . 8 ((𝑎 = 𝑦𝜑) → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
76alrimiv 1928 . . . . . . 7 ((𝑎 = 𝑦𝜑) → ∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
87expcom 417 . . . . . 6 (𝜑 → (𝑎 = 𝑦 → ∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦)))
98eximdv 1918 . . . . 5 (𝜑 → (∃𝑦 𝑎 = 𝑦 → ∃𝑦𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦)))
102, 9mpi 20 . . . 4 (𝜑 → ∃𝑦𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
11 ax6evr 2022 . . . . 5 𝑦 𝑏 = 𝑦
12 ifpfal 1072 . . . . . . . . . 10 𝜑 → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) ↔ 𝑧 = 𝑏))
1312biimpd 232 . . . . . . . . 9 𝜑 → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑏))
14 equtrr 2029 . . . . . . . . 9 (𝑏 = 𝑦 → (𝑧 = 𝑏𝑧 = 𝑦))
1513, 14sylan9r 512 . . . . . . . 8 ((𝑏 = 𝑦 ∧ ¬ 𝜑) → (if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
1615alrimiv 1928 . . . . . . 7 ((𝑏 = 𝑦 ∧ ¬ 𝜑) → ∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
1716expcom 417 . . . . . 6 𝜑 → (𝑏 = 𝑦 → ∀𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦)))
1817eximdv 1918 . . . . 5 𝜑 → (∃𝑦 𝑏 = 𝑦 → ∃𝑦𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦)))
1911, 18mpi 20 . . . 4 𝜑 → ∃𝑦𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
2010, 19pm2.61i 185 . . 3 𝑦𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦)
21 df-mo 2622 . . 3 (∃*𝑧if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) ↔ ∃𝑦𝑧(if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏) → 𝑧 = 𝑦))
2220, 21mpbir 234 . 2 ∃*𝑧if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏)
231, 22mpg 1799 1 𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  if-wif 1058  wal 1536  wex 1781  ∃*wmo 2620  wrex 3131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-rep 5166
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-ex 1782  df-mo 2622  df-rex 3136
This theorem is referenced by: (None)
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