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Theorem csbabgOLD 39552
Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) Obsolete as of 19-Aug-2018. Use csbab 4151 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbabgOLD (𝐴𝑉𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑})
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem csbabgOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sbccom 3650 . . . 4 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
2 df-clab 2747 . . . . 5 (𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
3 sbsbc 3580 . . . . 5 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
42, 3bitri 264 . . . 4 (𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
5 df-clab 2747 . . . . . 6 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
6 sbsbc 3580 . . . . . 6 ([𝑧 / 𝑦]𝜑[𝑧 / 𝑦]𝜑)
75, 6bitri 264 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
87sbcbii 3632 . . . 4 ([𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑} ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
91, 4, 83bitr4i 292 . . 3 (𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ [𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑})
10 sbcel2 4132 . . . 4 ([𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑} ↔ 𝑧𝐴 / 𝑥{𝑦𝜑})
1110a1i 11 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑} ↔ 𝑧𝐴 / 𝑥{𝑦𝜑}))
129, 11syl5rbb 273 . 2 (𝐴𝑉 → (𝑧𝐴 / 𝑥{𝑦𝜑} ↔ 𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑}))
1312eqrdv 2758 1 (𝐴𝑉𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  [wsb 2046  wcel 2139  {cab 2746  [wsbc 3576  csb 3674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-nul 4059
This theorem is referenced by:  csbingVD  39619  csbsngVD  39628  csbxpgVD  39629  csbrngVD  39631  csbunigVD  39633
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