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Theorem ceqex 2816
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 1570 . . 3 (𝑥 = 𝐴 → ∃𝑥 𝑥 = 𝐴)
2 isset 2695 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
31, 2sylibr 133 . 2 (𝑥 = 𝐴𝐴 ∈ V)
4 eqeq2 2150 . . . 4 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
54anbi1d 461 . . . . . 6 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
65exbidv 1798 . . . . 5 (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
76bibi2d 231 . . . 4 (𝑦 = 𝐴 → ((𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))))
84, 7imbi12d 233 . . 3 (𝑦 = 𝐴 → ((𝑥 = 𝑦 → (𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))) ↔ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))))
9 19.8a 1570 . . . . 5 ((𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
109ex 114 . . . 4 (𝑥 = 𝑦 → (𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)))
11 vex 2692 . . . . . 6 𝑦 ∈ V
1211alexeq 2815 . . . . 5 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜑))
13 sp 1489 . . . . . 6 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
1413com12 30 . . . . 5 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
1512, 14syl5bir 152 . . . 4 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑))
1610, 15impbid 128 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
178, 16vtoclg 2749 . 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 103  wb 104  wal 1330   = wceq 1332  wex 1469  wcel 1481  Vcvv 2689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691
This theorem is referenced by:  ceqsexg  2817  sbc6g  2937
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