Theorem elequ1 2171

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

Assertion

Ref	Expression
elequ1	⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))

Proof of Theorem elequ1

Step	Hyp	Ref	Expression
1		ax-13 2169	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
2		ax-13 2169	. . 3 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧))
3	2	equcoms 1722	. 2 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧))
4	1, 3	impbid 129	1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))

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 1463  ax-ie2 1508  ax-8 1518  ax-17 1540  ax-i9 1544  ax-13 2169

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  cleljust  2173  elsb1  2174  dveel1  2176  nalset  4163  zfpow  4208  mss  4259  zfun  4469  pw2f1odclem  6895  ctssdc  7179  acfun  7274  ccfunen  7331  bj-nalset  15541  bj-nnelirr  15599  nninfsellemqall  15659  nninfomni  15663