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  4254  nalset  4263  zfun  4579  fv3  5718  tfrlemisucaccv  6596  tfr1onlemsucaccv  6612  tfrcllemsucaccv  6625  sspw1or2  7544  acfun  7563  ccfunen  7630  cc1  7631  nninfinf  10893  bdsepnft  17011  bdsepnfALT  17013  bdbm1.3ii  17015  bj-nalset  17019  bj-nnelirr  17077  nninfalllem1  17149  nninfsellemeq  17155  nninfsellemqall  17156  nninfsellemeqinf  17157  nninfomni  17160