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Theorem equs5a 2451
Description: A property related to substitution that unlike equs5 2454 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2366. This proof uses ax12 2417, see equs5aALT 2358 for an alternative one using ax-12 2167 but not ax13 2369. Usage of the weaker equs5av 2266 is preferred, which uses ax12v2 2169, but not ax-13 2366. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
Assertion
Ref Expression
equs5a (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem equs5a
StepHypRef Expression
1 nfa1 2141 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
2 ax12 2417 . . 3 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
32imp 405 . 2 ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
41, 3exlimi 2206 1 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1532  wex 1774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-12 2167  ax-13 2366
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1775  df-nf 1779
This theorem is referenced by:  equs45f  2453
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