Theorem bnj1541 32123

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1541.1	⊢ (𝜑 ↔ (𝜓 ∧ 𝐴 ≠ 𝐵))
bnj1541.2	⊢ ¬ 𝜑

Assertion

Ref	Expression
bnj1541	⊢ (𝜓 → 𝐴 = 𝐵)

Proof of Theorem bnj1541

Step	Hyp	Ref	Expression
1		bnj1541.2	. . . 4 ⊢ ¬ 𝜑
2		bnj1541.1	. . . 4 ⊢ (𝜑 ↔ (𝜓 ∧ 𝐴 ≠ 𝐵))
3	1, 2	mtbi 324	. . 3 ⊢ ¬ (𝜓 ∧ 𝐴 ≠ 𝐵)
4	3	imnani 403	. 2 ⊢ (𝜓 → ¬ 𝐴 ≠ 𝐵)
5		nne 3020	. 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
6	4, 5	sylib 220	1 ⊢ (𝜓 → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ≠ wne 3016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 399  df-ne 3017

This theorem is referenced by:  bnj1312  32325  bnj1523  32338