Theorem eleq1w 2713

Description: Weaker version of eleq1 2718 (but more general than elequ1 2037) not depending on ax-ext 2631 nor df-cleq 2644. (Contributed by BJ, 24-Jun-2019.)

Assertion

Ref	Expression
eleq1w	⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))

Proof of Theorem eleq1w

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ2 1999	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑦))
2	1	anbi1d 741	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑧 = 𝑥 ∧ 𝑧 ∈ 𝐴) ↔ (𝑧 = 𝑦 ∧ 𝑧 ∈ 𝐴)))
3	2	exbidv 1890	. 2 ⊢ (𝑥 = 𝑦 → (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝐴) ↔ ∃𝑧(𝑧 = 𝑦 ∧ 𝑧 ∈ 𝐴)))
4		df-clel 2647	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝐴))
5		df-clel 2647	. 2 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑧(𝑧 = 𝑦 ∧ 𝑧 ∈ 𝐴))
6	3, 4, 5	3bitr4g 303	1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∃wex 1744   ∈ wcel 2030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-clel 2647

This theorem is referenced by:  reu8  3435  eqeuel  3974  reuccats1  13526  sumeven  15157  sumodd  15158  numedglnl  26084  uvtxnbgrvtx  26341  wspniunwspnon  26888  fusgr2wsp2nb  27314  numclwlk2lem2f1o  27359  numclwlk2lem2f1oOLD  27366  fsumiunle  29703  frpoinsg  31866  bj-clelsb3  32973  bj-nfcjust  32975  ftc1anclem6  33620  inxprnres  34201  monoordxr  40026  monoord2xr  40028  lmbr3  40297  cnrefiisp  40374  meaiunincf  41018  meaiuninc3v  41019  meaiuninc3  41020  sbgoldbm  41997