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Theorem ax9o 1588
 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 1586 . 2 𝑥 𝑥 = 𝑦
2 19.29r 1512 . . 3 ((∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑)) → ∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)))
3 hba1 1433 . . . . 5 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
4 pm3.35 329 . . . . 5 ((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → ∀𝑥𝜑)
53, 4exlimih 1484 . . . 4 (∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → ∀𝑥𝜑)
6 ax-4 1400 . . . 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 400 1 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241   = wceq 1243  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  equsalh  1614  spimth  1623  spimh  1625
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