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Theorem equs4v 2006
Description: Version of equs4 2438 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-May-1993.) (Revised 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 1972 . 2 𝑥 𝑥 = 𝑦
2 exintr 1893 . 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 398  wal 1535  wex 1780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-6 1970
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781
This theorem is referenced by:  sb1v  2095  sb2vOLDOLD  2512  sb2vOLDALT  2583  bj-sb56  34002  bj-equs45fv  34141
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