Theorem neleq12d 2303

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 2301	. . 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 2302	. . 3 ⊢ (𝐶 = 𝐷 → (𝐵 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷))
6	4, 5	syl 14	. 2 ⊢ (𝜑 → (𝐵 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷))
7	3, 6	bitrd 177	1 ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∉ wnel 2205

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036  df-nel 2207

This theorem is referenced by: (None)