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Theorem necon4ddc 2271
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
Hypothesis
Ref Expression
necon4ddc.1 (φ → (DECID A = B → (AB𝐶𝐷)))
Assertion
Ref Expression
necon4ddc (φ → (DECID A = B → (𝐶 = 𝐷A = B)))

Proof of Theorem necon4ddc
StepHypRef Expression
1 necon4ddc.1 . . 3 (φ → (DECID A = B → (AB𝐶𝐷)))
2 df-ne 2203 . . . 4 (AB ↔ ¬ A = B)
3 df-ne 2203 . . . 4 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
42, 3imbi12i 228 . . 3 ((AB𝐶𝐷) ↔ (¬ A = B → ¬ 𝐶 = 𝐷))
51, 4syl6ib 150 . 2 (φ → (DECID A = B → (¬ A = B → ¬ 𝐶 = 𝐷)))
6 condc 748 . 2 (DECID A = B → ((¬ A = B → ¬ 𝐶 = 𝐷) → (𝐶 = 𝐷A = B)))
75, 6sylcom 25 1 (φ → (DECID A = B → (𝐶 = 𝐷A = B)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  DECID wdc 741   = wceq 1242  wne 2201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 629
This theorem depends on definitions:  df-bi 110  df-dc 742  df-ne 2203
This theorem is referenced by: (None)
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