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Theorem bnj90 30762
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 5968 . . 3 (𝑥 = 𝑦 → (𝑧 Fn 𝑥𝑧 Fn 𝑦))
3 fneq2 5968 . . 3 (𝑦 = 𝑌 → (𝑧 Fn 𝑦𝑧 Fn 𝑌))
42, 3sbcie2g 3463 . 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 1988  Vcvv 3195  [wsbc 3429   Fn wfn 5871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-v 3197  df-sbc 3430  df-fn 5879
This theorem is referenced by:  bnj121  30914  bnj130  30918  bnj207  30925
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