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Theorem bnj90 31128
 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 6120 . . 3 (𝑥 = 𝑦 → (𝑧 Fn 𝑥𝑧 Fn 𝑦))
3 fneq2 6120 . . 3 (𝑦 = 𝑌 → (𝑧 Fn 𝑦𝑧 Fn 𝑌))
42, 3sbcie2g 3621 . 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 196   ∈ wcel 2145  Vcvv 3351  [wsbc 3587   Fn wfn 6026 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353  df-sbc 3588  df-fn 6034 This theorem is referenced by:  bnj121  31278  bnj130  31282  bnj207  31289
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