Theorem elequ1 2204

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 2202	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
2		ax-13 2202	. . 3 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧))
3	2	equcoms 1754	. 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 1495  ax-ie2 1540  ax-8 1550  ax-17 1572  ax-i9 1576  ax-13 2202

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  cleljust  2206  elsb1  2207  dveel1  2209  nalset  4213  zfpow  4258  mss  4311  zfun  4522  pw2f1odclem  6983  ctssdc  7268  acfun  7377  ccfunen  7438  bj-nalset  16188  bj-nnelirr  16246  2omap  16290  pw1map  16292  nninfsellemqall  16312  nninfomni  16316