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Theorem bnj62 33719
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj62 ([𝑧 / 𝑥]𝑥 Fn 𝐴𝑧 Fn 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐴(𝑧)

Proof of Theorem bnj62
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 3478 . . . 4 𝑦 ∈ V
2 fneq1 6637 . . . 4 (𝑥 = 𝑦 → (𝑥 Fn 𝐴𝑦 Fn 𝐴))
31, 2sbcie 3819 . . 3 ([𝑦 / 𝑥]𝑥 Fn 𝐴𝑦 Fn 𝐴)
43sbcbii 3836 . 2 ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴[𝑧 / 𝑦]𝑦 Fn 𝐴)
5 sbccow 3799 . 2 ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴[𝑧 / 𝑥]𝑥 Fn 𝐴)
6 vex 3478 . . 3 𝑧 ∈ V
7 fneq1 6637 . . 3 (𝑦 = 𝑧 → (𝑦 Fn 𝐴𝑧 Fn 𝐴))
86, 7sbcie 3819 . 2 ([𝑧 / 𝑦]𝑦 Fn 𝐴𝑧 Fn 𝐴)
94, 5, 83bitr3i 300 1 ([𝑧 / 𝑥]𝑥 Fn 𝐴𝑧 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  [wsbc 3776   Fn wfn 6535
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-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-fun 6542  df-fn 6543
This theorem is referenced by:  bnj156  33727  bnj976  33776  bnj581  33907
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