Theorem necon4addc 2437

Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)

Hypothesis

Ref	Expression
necon4addc.1	⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ¬ 𝜓)))

Assertion

Ref	Expression
necon4addc	⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓 → 𝐴 = 𝐵)))

Proof of Theorem necon4addc

Step	Hyp	Ref	Expression
1		necon4addc.1	. 2 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ¬ 𝜓)))
2		df-ne 2368	. . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	2	imbi1i 238	. . 3 ⊢ ((𝐴 ≠ 𝐵 → ¬ 𝜓) ↔ (¬ 𝐴 = 𝐵 → ¬ 𝜓))
4		condc 854	. . 3 ⊢ (DECID 𝐴 = 𝐵 → ((¬ 𝐴 = 𝐵 → ¬ 𝜓) → (𝜓 → 𝐴 = 𝐵)))
5	3, 4	biimtrid 152	. 2 ⊢ (DECID 𝐴 = 𝐵 → ((𝐴 ≠ 𝐵 → ¬ 𝜓) → (𝜓 → 𝐴 = 𝐵)))
6	1, 5	sylcom 28	1 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓 → 𝐴 = 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 835   = wceq 1364   ≠ wne 2367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-ne 2368

This theorem is referenced by:  4sqlem11  12570