Theorem bj-elequ12 33555

Description: An identity law for the non-logical predicate, which combines elequ1 2086 and elequ2 2094. For the analogous theorems for class terms, see eleq1 2868, eleq2 2869 and eleq12 2870. (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 2086	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
2		elequ2 2094	. 2 ⊢ (𝑧 = 𝑡 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡))
3	1, 2	sylan9bb 510	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 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1760

This theorem is referenced by:  bj-ru0  33770