Theorem elequ2 1641

Description: An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
elequ2	⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))

Proof of Theorem elequ2

Step	Hyp	Ref	Expression
1		ax-14 1445	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
2		ax-14 1445	. . 3 ⊢ (𝑦 = 𝑥 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥))
3	2	equcoms 1634	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥))
4	1, 3	impbid 127	1 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))

Syntax hints:   → wi 4   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1378  ax-ie2 1423  ax-8 1435  ax-14 1445  ax-17 1459  ax-i9 1463

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  elsb4  1894  dveel2  1938  axext3  2064  axext4  2065  bm1.1  2066  bm1.3ii  3899  nalset  3908  zfun  4189  fv3  5218  tfrlemisucaccv  5962  bdsepnft  10678  bdsepnfALT  10680  bdbm1.3ii  10682  bj-nalset  10686  bj-nnelirr  10748  strcollnft  10779  strcollnfALT  10781