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Theorem sbc5 2978
Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
Assertion
Ref Expression
sbc5 ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sbc5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbcex 2963 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
2 exsimpl 1610 . . 3 (∃𝑥(𝑥 = 𝐴𝜑) → ∃𝑥 𝑥 = 𝐴)
3 isset 2736 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
42, 3sylibr 133 . 2 (∃𝑥(𝑥 = 𝐴𝜑) → 𝐴 ∈ V)
5 dfsbcq2 2958 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
6 eqeq2 2180 . . . . 5 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
76anbi1d 462 . . . 4 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
87exbidv 1818 . . 3 (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
9 sb5 1880 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
105, 8, 9vtoclbg 2791 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
111, 4, 10pm5.21nii 699 1 ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1348  wex 1485  [wsb 1755  wcel 2141  Vcvv 2730  [wsbc 2955
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956
This theorem is referenced by:  sbc6g  2979  sbc7  2981  sbciegft  2985  sbccomlem  3029  csb2  3051  rexsns  3622
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