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Theorem bnj62 30771
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 3201 . . . 4 𝑦 ∈ V
2 fneq1 5977 . . . 4 (𝑥 = 𝑦 → (𝑥 Fn 𝐴𝑦 Fn 𝐴))
31, 2sbcie 3468 . . 3 ([𝑦 / 𝑥]𝑥 Fn 𝐴𝑦 Fn 𝐴)
43sbcbii 3489 . 2 ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴[𝑧 / 𝑦]𝑦 Fn 𝐴)
5 sbcco 3456 . 2 ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴[𝑧 / 𝑥]𝑥 Fn 𝐴)
6 vex 3201 . . 3 𝑧 ∈ V
7 fneq1 5977 . . 3 (𝑦 = 𝑧 → (𝑦 Fn 𝐴𝑧 Fn 𝐴))
86, 7sbcie 3468 . 2 ([𝑧 / 𝑦]𝑦 Fn 𝐴𝑧 Fn 𝐴)
94, 5, 83bitr3i 290 1 ([𝑧 / 𝑥]𝑥 Fn 𝐴𝑧 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsbc 3433   Fn wfn 5881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-fun 5888  df-fn 5889
This theorem is referenced by:  bnj156  30781  bnj976  30833  bnj581  30963
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