Theorem bj-elequ12 34787

Description: An identity law for the non-logical predicate, which combines elequ1 2115 and elequ2 2123. For the analogous theorems for class terms, see eleq1 2826, eleq2 2827 and eleq12 2828. TODO: move to main part. (Contributed by BJ, 29-Sep-2019.)

Assertion

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

Proof of Theorem bj-elequ12

Step	Hyp	Ref	Expression
1		elequ1 2115	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
2		elequ2 2123	. 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 205   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784

This theorem is referenced by:  bj-ru0  35058