Theorem elequ1 2209

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 2207	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
2		ax-13 2207	. . 3 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧))
3	2	equcoms 1756	. 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 1498  ax-ie2 1543  ax-8 1553  ax-17 1575  ax-i9 1579  ax-13 2207

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  cleljust  2211  elsb1  2212  dveel1  2214  nalset  4245  zfpow  4293  mss  4347  zfun  4560  pw2f1odclem  7100  2omap  7282  ctssdc  7417  acfun  7527  ccfunen  7594  hashfibclem  11231  bj-nalset  16777  bj-nnelirr  16835  pw1map  16881  nninfsellemqall  16905  nninfomni  16909