Theorem eleq2w 2819

Description: Weaker version of eleq2 2824 (but more general than elequ2 2129) not depending on ax-ext 2707 nor df-cleq 2727. (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 2129	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
2	1	anbi2d 631	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑥) ↔ (𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑦)))
3	2	exbidv 1923	. 2 ⊢ (𝑥 = 𝑦 → (∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑥) ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑦)))
4		dfclel 2811	. 2 ⊢ (𝐴 ∈ 𝑥 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑥))
5		dfclel 2811	. 2 ⊢ (𝐴 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝑦))
6	3, 4, 5	3bitr4g 314	1 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-clel 2810

This theorem is referenced by:  clelsb2  2863  eluniab  4876  elintabg  4912  cantnflem1c  9598  tcrank  9798  isf32lem2  10266  sadcp1  16384  subgacs  19092  nsgacs  19093  sdrgacs  20736  lssacs  20920  elcls3  23029  conncompconn  23378  1stcfb  23391  dfac14lem  23563  r0cld  23684  uffix  23867  flftg  23942  tgpconncompeqg  24058  wilth  27039  tghilberti2  28691  umgr2edgneu  29268  uspgredg2v  29278  usgredgleordALT  29288  nbusgrf1o  29425  vtxdushgrfvedglem  29544  constrmon  33880  ddemeas  34372  cvmcov  35436  cvmseu  35449  sat1el2xp  35552  hilbert1.2  36328  fneint  36521  mnuprdlem1  44550  mnuprdlem2  44551  mnuprdlem4  44553  elunif  45298  fnchoice  45311  lmbr3  46028