Theorem bnj1541 30180

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 311	. . 3 ⊢ ¬ (𝜓 ∧ 𝐴 ≠ 𝐵)
4	3	imnani 438	. 2 ⊢ (𝜓 → ¬ 𝐴 ≠ 𝐵)
5		nne 2786	. 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
6	4, 5	sylib 207	1 ⊢ (𝜓 → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ≠ wne 2780

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 196  df-an 385  df-ne 2782

This theorem is referenced by:  bnj1312  30380  bnj1523  30393