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Theorem ceqex 2731
Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
Assertion
Ref Expression
ceqex (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 19.8a 1523 . . 3 (𝑥 = 𝐴 → ∃𝑥 𝑥 = 𝐴)
2 isset 2615 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
31, 2sylibr 132 . 2 (𝑥 = 𝐴𝐴 ∈ V)
4 eqeq2 2092 . . . 4 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
54anbi1d 453 . . . . . 6 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
65exbidv 1748 . . . . 5 (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
76bibi2d 230 . . . 4 (𝑦 = 𝐴 → ((𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))))
84, 7imbi12d 232 . . 3 (𝑦 = 𝐴 → ((𝑥 = 𝑦 → (𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))) ↔ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))))
9 19.8a 1523 . . . . 5 ((𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
109ex 113 . . . 4 (𝑥 = 𝑦 → (𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)))
11 vex 2614 . . . . . 6 𝑦 ∈ V
1211alexeq 2730 . . . . 5 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜑))
13 sp 1442 . . . . . 6 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
1413com12 30 . . . . 5 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
1512, 14syl5bir 151 . . . 4 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑))
1610, 15impbid 127 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
178, 16vtoclg 2668 . 2 (𝐴 ∈ V → (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))))
183, 17mpcom 36 1 (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wex 1422  wcel 1434  Vcvv 2611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2613
This theorem is referenced by:  ceqsexg  2732  sbc6g  2849
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