Theorem elequ12 2137

Description: An identity law for the non-logical predicate, which combines elequ1 2126 and elequ2 2134. The analogous theorems for class terms are eleq1 2828, eleq2 2829, and eleq12 2830 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 2126	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
2		elequ2 2134	. 2 ⊢ (𝑧 = 𝑡 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡))
3	1, 2	sylan9bb 514	1 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787

This theorem is referenced by:  ru0  2138  elirrv  9509  axtcond  36713  mh-inf3f1  36776  mh-unprimbi  36779  fvineqsneu  37780