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Theorem bj-aecomsv 33074
 Description: Version of aecoms 2454 with a dv condition, provable from Tarski's FOL. The corresponding version of naecoms 2455 should not be very useful since ¬ ∀𝑥𝑥 = 𝑦, DV(x, y) is true when the universe has at least two objects (see bj-dtru 33125). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-aecomsv.1 (∀𝑥 𝑥 = 𝑦𝜑)
Assertion
Ref Expression
bj-aecomsv (∀𝑦 𝑦 = 𝑥𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-aecomsv
StepHypRef Expression
1 bj-axc11nv 33073 . 2 (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑥 = 𝑦)
2 bj-aecomsv.1 . 2 (∀𝑥 𝑥 = 𝑦𝜑)
31, 2syl 17 1 (∀𝑦 𝑦 = 𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1630 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854 This theorem is referenced by:  bj-axc11v  33075
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