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Theorem ax11v2 1742
Description: Recovery of ax11o 1744 from ax11v 1749 without using ax-11 1438. The hypothesis is even weaker than ax11v 1749, with 𝑧 both distinct from 𝑥 and not occurring in 𝜑. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1744. (Contributed by NM, 2-Feb-2007.)
Hypothesis
Ref Expression
ax11v2.1 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
Assertion
Ref Expression
ax11v2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax11v2
StepHypRef Expression
1 a9e 1627 . 2 𝑧 𝑧 = 𝑦
2 ax11v2.1 . . . . 5 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
3 equequ2 1640 . . . . . . 7 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
43adantl 271 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → (𝑥 = 𝑧𝑥 = 𝑦))
5 dveeq2 1737 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
65imp 122 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → ∀𝑥 𝑧 = 𝑦)
7 hba1 1474 . . . . . . . . 9 (∀𝑥 𝑧 = 𝑦 → ∀𝑥𝑥 𝑧 = 𝑦)
83imbi1d 229 . . . . . . . . . 10 (𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
98sps 1471 . . . . . . . . 9 (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
107, 9albidh 1410 . . . . . . . 8 (∀𝑥 𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
116, 10syl 14 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → (∀𝑥(𝑥 = 𝑧𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
1211imbi2d 228 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → ((𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
134, 12imbi12d 232 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑))) ↔ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
142, 13mpbii 146 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
1514ex 113 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
1615exlimdv 1741 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
171, 16mpi 15 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wal 1283  wex 1422
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-in2 578  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-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by:  ax11a2  1743
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