Theorem neleq12d 2609

Description: Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)

Hypotheses

Ref	Expression
neleq12d.1	⊢ (φ → A = B)
neleq12d.2	⊢ (φ → C = D)

Assertion

Ref	Expression
neleq12d	⊢ (φ → (A ∉ C ↔ B ∉ D))

Proof of Theorem neleq12d

Step	Hyp	Ref	Expression
1		neleq12d.1	. . 3 ⊢ (φ → A = B)
2		neleq1 2607	. . 3 ⊢ (A = B → (A ∉ C ↔ B ∉ C))
3	1, 2	syl 15	. 2 ⊢ (φ → (A ∉ C ↔ B ∉ C))
4		neleq12d.2	. . 3 ⊢ (φ → C = D)
5		neleq2 2608	. . 3 ⊢ (C = D → (B ∉ C ↔ B ∉ D))
6	4, 5	syl 15	. 2 ⊢ (φ → (B ∉ C ↔ B ∉ D))
7	3, 6	bitrd 244	1 ⊢ (φ → (A ∉ C ↔ B ∉ D))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∉ wnel 2517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-nel 2519

This theorem is referenced by: (None)