Theorem elequ2 2214

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 2212	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
2		ax-14 2212	. . 3 ⊢ (𝑦 = 𝑥 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥))
3	2	equcoms 1760	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥))
4	1, 3	impbid 129	1 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1502  ax-ie2 1547  ax-8 1557  ax-17 1579  ax-i9 1583  ax-14 2212

This proof depends on definitions:  df-bi 117

This theorem is used by:  elsb2  2217  dveel2  2219  axext3  2221  axext4  2222  bm1.1  2223  eleq2w  2300  bm1.3ii  4256  nalset  4265  zfun  4581  fv3  5721  tfrlemisucaccv  6601  tfr1onlemsucaccv  6617  tfrcllemsucaccv  6630  sspw1or2  7550  acfun  7569  ccfunen  7636  cc1  7637  nninfinf  10900  bdsepnft  17135  bdsepnfALT  17137  bdbm1.3ii  17139  bj-nalset  17143  bj-nnelirr  17201  nninfalllem1  17274  nninfsellemeq  17280  nninfsellemqall  17281  nninfsellemeqinf  17282  nninfomni  17285