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Theorem equs4v 1976
 Description: Version of equs4 2326 with a dv condition, which requires fewer axioms. (Contributed by BJ, 31-May-2019.)
Assertion
Ref Expression
equs4v (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem equs4v
StepHypRef Expression
1 ax6ev 1947 . 2 𝑥 𝑥 = 𝑦
2 exintr 1859 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑦𝜑)))
31, 2mpi 20 1 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1521  ∃wex 1744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-6 1945 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745 This theorem is referenced by:  equvelv  2005  bj-sb56  32764  bj-equs45fv  32877  bj-sb2v  32878
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