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Theorem bnj90 32220
 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 6426 . . 3 (𝑥 = 𝑦 → (𝑧 Fn 𝑥𝑧 Fn 𝑦))
3 fneq2 6426 . . 3 (𝑦 = 𝑌 → (𝑧 Fn 𝑦𝑧 Fn 𝑌))
42, 3sbcie2g 3736 . 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 209   ∈ wcel 2111  Vcvv 3409  [wsbc 3696   Fn wfn 6330 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 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-sbc 3697  df-fn 6338 This theorem is referenced by:  bnj121  32370  bnj130  32374  bnj207  32381
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