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Theorem equs5 2477
Description: Lemma used in proofs of substitution properties. If there is a disjoint variable condition on 𝑥, 𝑦, then sb56 2271 can be used instead; if 𝑦 is not free in 𝜑, then equs45f 2476 can be used. Usage of this theorem is discouraged because it depends on ax-13 2384. (Contributed by NM, 14-May-1993.) (Revised by BJ, 1-Oct-2018.) (New usage is discouraged.)
Assertion
Ref Expression
equs5 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem equs5
StepHypRef Expression
1 nfna1 2150 . . 3 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2 nfa1 2149 . . 3 𝑥𝑥(𝑥 = 𝑦𝜑)
3 axc15 2438 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
43impd 413 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
51, 2, 4exlimd 2211 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
6 equs4 2432 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
75, 6impbid1 227 1 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1529  wex 1774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-10 2139  ax-12 2170  ax-13 2384
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1775  df-nf 1779
This theorem is referenced by:  sb3b  2495  sb3OLD  2499  sb4ALT  2582  bj-sbsb  34153
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