Theorem eqabdv 2878

Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) Avoid ax-11 2158. (Revised by Wolf Lammen, 6-May-2023.)

Hypothesis

Ref	Expression
eqabdv.1	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))

Assertion

Ref	Expression
eqabdv	⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem eqabdv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqabdv.1	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))
2	1	sbbidv 2079	. . 3 ⊢ (𝜑 → ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝜓))
3		clelsb1 2871	. . . 4 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
4	3	bicomi 224	. . 3 ⊢ (𝑦 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴)
5		df-clab 2718	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓)
6	2, 4, 5	3bitr4g 314	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ {𝑥 ∣ 𝜓}))
7	6	eqrdv 2738	1 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537  [wsb 2064   ∈ wcel 2108  {cab 2717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819

This theorem is referenced by:  eqabcdv  2879  eqabi  2880  sbab  2892  rabeqcda  3455  iftrue  4554  iffalse  4557  dfopif  4894  iniseg  6127  setlikespec  6357  fncnvima2  7094  isoini  7374  dftpos3  8285  mapsnd  8944  hartogslem1  9611  r1val2  9906  cardval2  10060  dfac3  10190  wrdval  14565  wrdnval  14593  submgmacs  18755  submacs  18862  ablsimpgfind  20154  dfrhm2  20500  lsppr  21115  rspsn  21366  znunithash  21606  tgval3  22991  txrest  23660  xkoptsub  23683  cnextf  24095  cnblcld  24816  shft2rab  25562  sca2rab  25566  renegscl  28448  grpoinvf  30564  elpjrn  32222  ofrn2  32659  ellcsrspsn  35609  neibastop3  36328  ecres2  38235  lkrval2  39046  lshpset2N  39075  hdmapoc  41888