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Theorem sbcrexgOLD 42773
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 3884 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 3794 . 2 (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃𝑦𝐵 𝜑[𝐴 / 𝑥]𝑦𝐵 𝜑))
2 dfsbcq2 3794 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32rexbidv 3177 . 2 (𝑧 = 𝐴 → (∃𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
4 nfcv 2903 . . . 4 𝑥𝐵
5 nfs1v 2154 . . . 4 𝑥[𝑧 / 𝑥]𝜑
64, 5nfrexw 3311 . . 3 𝑥𝑦𝐵 [𝑧 / 𝑥]𝜑
7 sbequ12 2249 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
87rexbidv 3177 . . 3 (𝑥 = 𝑧 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑))
96, 8sbie 2505 . 2 ([𝑧 / 𝑥]∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
101, 3, 9vtoclbg 3557 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  [wsb 2062  wcel 2106  wrex 3068  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-sbc 3792
This theorem is referenced by:  2sbcrexOLD  42774  sbc2rexgOLD  42776
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