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Theorem equs5 2338
Description: Lemma used in proofs of substitution properties. If there is a dv condition on 𝑥, 𝑦, then sb56 2134 can be used instead; if 𝑦 is not free in 𝜑, then equs45f 2337 can be used. (Contributed by NM, 14-May-1993.) (Revised by BJ, 1-Oct-2018.)
Assertion
Ref Expression
equs5 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem equs5
StepHypRef Expression
1 nfna1 2015 . . 3 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2 nfa1 2014 . . 3 𝑥𝑥(𝑥 = 𝑦𝜑)
3 axc15 2290 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
43impd 445 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
51, 2, 4exlimd 2072 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
6 equs4 2276 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
75, 6impbid1 213 1 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2032  ax-13 2232
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700
This theorem is referenced by:  sb3  2342  sb4  2343  bj-sbsb  31825
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