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Theorem equs5a 2437
 Description: A property related to substitution that unlike equs5 2440 does not require a distinctor antecedent. See equs5aALT 2341 for an alternate proof using ax-12 2141 but not ax13 2347. (Contributed by NM, 2-Feb-2007.)
Assertion
Ref Expression
equs5a (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem equs5a
StepHypRef Expression
1 nfa1 2121 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
2 ax12 2402 . . 3 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
32imp 407 . 2 ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
41, 3exlimi 2182 1 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1520  ∃wex 1761 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141  ax-13 2344 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766 This theorem is referenced by:  equs45f  2439  sb4aALT  2552  bj-equs45fv  33683
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