Theorem elequ12 2129

Description: An identity law for the non-logical predicate, which combines elequ1 2118 and elequ2 2126. The analogous theorems for class terms are eleq1 2819, eleq2 2820, and eleq12 2821 respectively. (Contributed by BJ, 29-Sep-2019.)

Assertion

Ref	Expression
elequ12	⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡))

Proof of Theorem elequ12

Step	Hyp	Ref	Expression
1		elequ1 2118	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
2		elequ2 2126	. 2 ⊢ (𝑧 = 𝑡 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡))
3	1, 2	sylan9bb 509	1 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781

This theorem is referenced by:  ru0  2130  fvineqsneu  37444