Theorem bj-elequ12 33005

Description: An identity law for the non-logical predicate, which combines elequ1 2152 and elequ2 2159. For the analogous theorems for class terms, see eleq1 2838, eleq2 2839 and eleq12 2840. (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 2152	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
2		elequ2 2159	. 2 ⊢ (𝑧 = 𝑡 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡))
3	1, 2	sylan9bb 499	1 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154

This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853

This theorem is referenced by:  bj-ru0  33264