Theorem eleq2w 2226

Description: Weaker version of eleq2 2228 (but more general than elequ2 2140) not depending on ax-ext 2146 nor df-cleq 2157. (Contributed by BJ, 29-Sep-2019.)

Assertion

Ref	Expression
eleq2w	⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))

Proof of Theorem eleq2w

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ2 2140	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
2	1	anbi2d 460	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑥) ↔ (𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑦)))
3	2	exbidv 1812	. 2 ⊢ (𝑥 = 𝑦 → (∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑥) ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑦)))
4		df-clel 2160	. 2 ⊢ (𝐴 ∈ 𝑥 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑥))
5		df-clel 2160	. 2 ⊢ (𝐴 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑦))
6	3, 4, 5	3bitr4g 222	1 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1342  ∃wex 1479   ∈ wcel 2135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-14 2138

This theorem depends on definitions:  df-bi 116  df-clel 2160

This theorem is referenced by:  exmidontriimlem4  7171