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Theorem bnj90 31879
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj90.1 𝑌 ∈ V
Assertion
Ref Expression
bnj90 ([𝑌 / 𝑥]𝑧 Fn 𝑥𝑧 Fn 𝑌)
Distinct variable group:   𝑥,𝑧
Allowed substitution hints:   𝑌(𝑥,𝑧)

Proof of Theorem bnj90
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 bnj90.1 . 2 𝑌 ∈ V
2 fneq2 6441 . . 3 (𝑥 = 𝑦 → (𝑧 Fn 𝑥𝑧 Fn 𝑦))
3 fneq2 6441 . . 3 (𝑦 = 𝑌 → (𝑧 Fn 𝑦𝑧 Fn 𝑌))
42, 3sbcie2g 3814 . 2 (𝑌 ∈ V → ([𝑌 / 𝑥]𝑧 Fn 𝑥𝑧 Fn 𝑌))
51, 4ax-mp 5 1 ([𝑌 / 𝑥]𝑧 Fn 𝑥𝑧 Fn 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wcel 2106  Vcvv 3499  [wsbc 3775   Fn wfn 6346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-sbc 3776  df-fn 6354
This theorem is referenced by:  bnj121  32029  bnj130  32033  bnj207  32040
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