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Theorem sbc5ALT 3768
Description: Alternate proof of sbc5 3767. This proof helps show how clelab 2883 works, since it is equivalent but shorter thanks to now-available library theorems like vtoclbg 3528 and isset 3458. (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 3749 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
2 exsimpl 1871 . . 3 (∃𝑥(𝑥 = 𝐴𝜑) → ∃𝑥 𝑥 = 𝐴)
3 isset 3458 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
42, 3sylibr 233 . 2 (∃𝑥(𝑥 = 𝐴𝜑) → 𝐴 ∈ V)
5 dfsbcq2 3742 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
6 eqeq2 2748 . . . . 5 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
76anbi1d 630 . . . 4 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
87exbidv 1924 . . 3 (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
9 sb5 2267 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
105, 8, 9vtoclbg 3528 . 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 396   = wceq 1541  wex 1781  [wsb 2067  wcel 2106  Vcvv 3445  [wsbc 3739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3447  df-sbc 3740
This theorem is referenced by: (None)
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