Theorem neleq12d 2521

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 2519	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶))
4		neleq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
5		neleq2 2520	. . 3 ⊢ (𝐶 = 𝐷 → (𝐵 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷))
6	4, 5	syl 14	. 2 ⊢ (𝜑 → (𝐵 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷))
7	3, 6	bitrd 188	1 ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1402   ∉ wnel 2515

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 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234  df-nel 2516

This theorem is referenced by: (None)