Theorem elequ2 1601

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 1405	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
2		ax-14 1405	. . 3 ⊢ (𝑦 = 𝑥 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥))
3	2	equcoms 1594	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥))
4	1, 3	impbid 120	1 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))

Syntax hints:   → wi 4   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338  ax-ie2 1383  ax-8 1395  ax-14 1405  ax-17 1419  ax-i9 1423

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  elsb4  1853  dveel2  1897  axext3  2023  axext4  2024  bm1.1  2025  bm1.3ii  3878  nalset  3887  zfun  4171  fv3  5197  tfrlemisucaccv  5939  bdsepnft  10007  bdsepnfALT  10009  bdbm1.3ii  10011  bj-nalset  10015  bj-nnelirr  10078  strcollnft  10109  strcollnfALT  10111