Theorem elequ12 2159

Description: An identity law for the non-logical predicate, which combines elequ1 2148 and elequ2 2156. The analogous theorems for class terms are eleq1 2849, eleq2 2850, and eleq12 2851 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 2148	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
2		elequ2 2156	. 2 ⊢ (𝑧 = 𝑡 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡))
3	1, 2	sylan9bb 517	1 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799

This theorem is referenced by:  ru0  2160  elirrv  9542  axtcond  36802  mh-inf3f1  36865  mh-unprimbi  36868  fvineqsneu  37869