Theorem elequ2 2153

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 2151	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
2		ax-14 2151	. . 3 ⊢ (𝑦 = 𝑥 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥))
3	2	equcoms 1708	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥))
4	1, 3	impbid 129	1 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1449  ax-ie2 1494  ax-8 1504  ax-17 1526  ax-i9 1530  ax-14 2151

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  elsb2  2156  dveel2  2158  axext3  2160  axext4  2161  bm1.1  2162  eleq2w  2239  bm1.3ii  4126  nalset  4135  zfun  4436  fv3  5540  tfrlemisucaccv  6328  tfr1onlemsucaccv  6344  tfrcllemsucaccv  6357  acfun  7208  ccfunen  7265  cc1  7266  bdsepnft  14724  bdsepnfALT  14726  bdbm1.3ii  14728  bj-nalset  14732  bj-nnelirr  14790  nninfalllem1  14842  nninfsellemeq  14848  nninfsellemqall  14849  nninfsellemeqinf  14850  nninfomni  14853