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Theorem equidqe 35500
 Description: equid 1969 with existential quantifier without using ax-c5 35461 or ax-5 1869. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equidqe ¬ ∀𝑦 ¬ 𝑥 = 𝑥

Proof of Theorem equidqe
StepHypRef Expression
1 ax6fromc10 35474 . 2 ¬ ∀𝑦 ¬ 𝑦 = 𝑥
2 ax7 1973 . . . . 5 (𝑦 = 𝑥 → (𝑦 = 𝑥𝑥 = 𝑥))
32pm2.43i 52 . . . 4 (𝑦 = 𝑥𝑥 = 𝑥)
43con3i 152 . . 3 𝑥 = 𝑥 → ¬ 𝑦 = 𝑥)
54alimi 1774 . 2 (∀𝑦 ¬ 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑦 = 𝑥)
61, 5mto 189 1 ¬ ∀𝑦 ¬ 𝑥 = 𝑥
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3  ∀wal 1505 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-c7 35463  ax-c10 35464 This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743 This theorem is referenced by:  axc5sp1  35501  equidq  35502
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