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Theorem sbcrexgOLD 37666
 Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcrex 3547 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcrexgOLD (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcrexgOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3471 . 2 (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃𝑦𝐵 𝜑[𝐴 / 𝑥]𝑦𝐵 𝜑))
2 dfsbcq2 3471 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32rexbidv 3081 . 2 (𝑧 = 𝐴 → (∃𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
4 nfcv 2793 . . . 4 𝑥𝐵
5 nfs1v 2465 . . . 4 𝑥[𝑧 / 𝑥]𝜑
64, 5nfrex 3036 . . 3 𝑥𝑦𝐵 [𝑧 / 𝑥]𝜑
7 sbequ12 2149 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
87rexbidv 3081 . . 3 (𝑥 = 𝑧 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑))
96, 8sbie 2436 . 2 ([𝑧 / 𝑥]∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
101, 3, 9vtoclbg 3298 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1523  [wsb 1937   ∈ wcel 2030  ∃wrex 2942  [wsbc 3468 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-sbc 3469 This theorem is referenced by:  2sbcrexOLD  37667  sbc2rexgOLD  37669
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