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Theorem equs5aALT 2310
Description: Alternate proof of equs5a 2473. Uses ax-12 2184 but not ax-13 2379. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equs5aALT (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem equs5aALT
StepHypRef Expression
1 nfa1 2165 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
2 ax-12 2184 . . 3 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
32imp 444 . 2 ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
41, 3exlimi 2221 1 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1618  wex 1841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-10 2156  ax-12 2184
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1842  df-nf 1847
This theorem is referenced by: (None)
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