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Theorem bnj62 32104
 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 3447 . . . 4 𝑦 ∈ V
2 fneq1 6418 . . . 4 (𝑥 = 𝑦 → (𝑥 Fn 𝐴𝑦 Fn 𝐴))
31, 2sbcie 3763 . . 3 ([𝑦 / 𝑥]𝑥 Fn 𝐴𝑦 Fn 𝐴)
43sbcbii 3779 . 2 ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴[𝑧 / 𝑦]𝑦 Fn 𝐴)
5 sbccow 3746 . 2 ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴[𝑧 / 𝑥]𝑥 Fn 𝐴)
6 vex 3447 . . 3 𝑧 ∈ V
7 fneq1 6418 . . 3 (𝑦 = 𝑧 → (𝑦 Fn 𝐴𝑧 Fn 𝐴))
86, 7sbcie 3763 . 2 ([𝑧 / 𝑦]𝑦 Fn 𝐴𝑧 Fn 𝐴)
94, 5, 83bitr3i 304 1 ([𝑧 / 𝑥]𝑥 Fn 𝐴𝑧 Fn 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  [wsbc 3723   Fn wfn 6323 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-sbc 3724  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-fun 6330  df-fn 6331 This theorem is referenced by:  bnj156  32112  bnj976  32163  bnj581  32294
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