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Theorem ceqex 2693
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 1498 . . 3 (𝑥 = 𝐴 → ∃𝑥 𝑥 = 𝐴)
2 isset 2578 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
31, 2sylibr 141 . 2 (𝑥 = 𝐴𝐴 ∈ V)
4 eqeq2 2065 . . . 4 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
54anbi1d 446 . . . . . 6 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
65exbidv 1722 . . . . 5 (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
76bibi2d 225 . . . 4 (𝑦 = 𝐴 → ((𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))))
84, 7imbi12d 227 . . 3 (𝑦 = 𝐴 → ((𝑥 = 𝑦 → (𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))) ↔ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))))
9 19.8a 1498 . . . . 5 ((𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
109ex 112 . . . 4 (𝑥 = 𝑦 → (𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)))
11 vex 2577 . . . . . 6 𝑦 ∈ V
1211alexeq 2692 . . . . 5 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜑))
13 sp 1417 . . . . . 6 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
1413com12 30 . . . . 5 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
1512, 14syl5bir 146 . . . 4 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑))
1610, 15impbid 124 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
178, 16vtoclg 2630 . 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 101  wb 102  wal 1257   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  ceqsexg  2694  sbc6g  2810
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