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Theorem equs5a 2456
Description: A property related to substitution that unlike equs5 2459 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2371. This proof uses ax12 2422, see equs5aALT 2365 for an alternative one using ax-12 2175 but not ax13 2374. Usage of the weaker equs5av 2275 is preferred, which uses ax12v2 2177, but not ax-13 2371. (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 2152 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
2 ax12 2422 . . 3 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
32imp 410 . 2 ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
41, 3exlimi 2215 1 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1541  wex 1787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-12 2175  ax-13 2371
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-nf 1792
This theorem is referenced by:  equs45f  2458
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