Theorem eleq1w 2684

Description: Weaker version of eleq1 2689 (but more general than elequ1 1997) not depending on ax-ext 2602 (nor ax-12 2047 nor df-cleq 2615). (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 1953	. . . 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 1850	. 2 ⊢ (𝑥 = 𝑦 → (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝐴) ↔ ∃𝑧(𝑧 = 𝑦 ∧ 𝑧 ∈ 𝐴)))
4		df-clel 2618	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝐴))
5		df-clel 2618	. 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 384  ∃wex 1704   ∈ wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-clel 2618

This theorem is referenced by:  reu8  3402  eqeuel  3941  reuccats1  13480  sumeven  15110  sumodd  15111  numedglnl  26039  fusgr2wsp2nb  27198  numclwlk2lem2f1o  27238  fsumiunle  29575  bj-clelsb3  32848  bj-nfcjust  32850  ftc1anclem6  33490  inxprnres  34060  lmbr3  39979  cnrefiisp  40056  sbgoldbm  41672