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Theorem ax9o 1698
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 1696 . 2 𝑥 𝑥 = 𝑦
2 19.29r 1621 . . 3 ((∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑)) → ∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)))
3 hba1 1540 . . . . 5 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
4 pm3.35 347 . . . . 5 ((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → ∀𝑥𝜑)
53, 4exlimih 1593 . . . 4 (∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → ∀𝑥𝜑)
6 ax-4 1510 . . . 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 424 1 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1351   = wceq 1353  wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  equsalh  1726  spimth  1735  spimh  1737
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