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Theorem ax9o 1629
Description: An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
Assertion
Ref Expression
ax9o (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Proof of Theorem ax9o
StepHypRef Expression
1 a9e 1627 . 2 𝑥 𝑥 = 𝑦
2 19.29r 1553 . . 3 ((∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑)) → ∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)))
3 hba1 1474 . . . . 5 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
4 pm3.35 339 . . . . 5 ((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → ∀𝑥𝜑)
53, 4exlimih 1525 . . . 4 (∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → ∀𝑥𝜑)
6 ax-4 1441 . . . 4 (∀𝑥𝜑𝜑)
75, 6syl 14 . . 3 (∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → 𝜑)
82, 7syl 14 . 2 ((∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑)) → 𝜑)
91, 8mpan 415 1 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1283   = wceq 1285  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-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  equsalh  1655  spimth  1664  spimh  1666
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