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Theorem ceqex 2864
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 1590 . . 3 (𝑥 = 𝐴 → ∃𝑥 𝑥 = 𝐴)
2 isset 2743 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
31, 2sylibr 134 . 2 (𝑥 = 𝐴𝐴 ∈ V)
4 eqeq2 2187 . . . 4 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
54anbi1d 465 . . . . . 6 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
65exbidv 1825 . . . . 5 (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
76bibi2d 232 . . . 4 (𝑦 = 𝐴 → ((𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))))
84, 7imbi12d 234 . . 3 (𝑦 = 𝐴 → ((𝑥 = 𝑦 → (𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))) ↔ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))))
9 19.8a 1590 . . . . 5 ((𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
109ex 115 . . . 4 (𝑥 = 𝑦 → (𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)))
11 vex 2740 . . . . . 6 𝑦 ∈ V
1211alexeq 2863 . . . . 5 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜑))
13 sp 1511 . . . . . 6 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
1413com12 30 . . . . 5 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
1512, 14biimtrrid 153 . . . 4 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑))
1610, 15impbid 129 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
178, 16vtoclg 2797 . 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 104  wb 105  wal 1351   = wceq 1353  wex 1492  wcel 2148  Vcvv 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739
This theorem is referenced by:  ceqsexg  2865  sbc6g  2987
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