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Theorem sbc5ALT 3747
Description: Alternate proof of sbc5 3746. This proof helps show how clelab 2878 works, since it is equivalent but shorter thanks to now-available library theorems like vtoclbg 3509 and isset 3447. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbc5ALT ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sbc5ALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbcex 3728 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
2 exsimpl 1867 . . 3 (∃𝑥(𝑥 = 𝐴𝜑) → ∃𝑥 𝑥 = 𝐴)
3 isset 3447 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
42, 3sylibr 233 . 2 (∃𝑥(𝑥 = 𝐴𝜑) → 𝐴 ∈ V)
5 dfsbcq2 3721 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
6 eqeq2 2745 . . . . 5 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
76anbi1d 629 . . . 4 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
87exbidv 1920 . . 3 (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
9 sb5 2263 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
105, 8, 9vtoclbg 3509 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
111, 4, 10pm5.21nii 379 1 ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1537  wex 1777  [wsb 2062  wcel 2101  Vcvv 3434  [wsbc 3718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-12 2166  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3436  df-sbc 3719
This theorem is referenced by: (None)
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