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Theorem bnj156 31024
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj156.1 (𝜁0 ↔ (𝑓 Fn 1𝑜𝜑′𝜓′))
bnj156.2 (𝜁1[𝑔 / 𝑓]𝜁0)
bnj156.3 (𝜑1[𝑔 / 𝑓]𝜑′)
bnj156.4 (𝜓1[𝑔 / 𝑓]𝜓′)
Assertion
Ref Expression
bnj156 (𝜁1 ↔ (𝑔 Fn 1𝑜𝜑1𝜓1))

Proof of Theorem bnj156
StepHypRef Expression
1 bnj156.2 . 2 (𝜁1[𝑔 / 𝑓]𝜁0)
2 bnj156.1 . . . 4 (𝜁0 ↔ (𝑓 Fn 1𝑜𝜑′𝜓′))
32sbcbii 3597 . . 3 ([𝑔 / 𝑓]𝜁0[𝑔 / 𝑓](𝑓 Fn 1𝑜𝜑′𝜓′))
4 sbc3an 3600 . . . 4 ([𝑔 / 𝑓](𝑓 Fn 1𝑜𝜑′𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 1𝑜[𝑔 / 𝑓]𝜑′[𝑔 / 𝑓]𝜓′))
5 bnj62 31016 . . . . 5 ([𝑔 / 𝑓]𝑓 Fn 1𝑜𝑔 Fn 1𝑜)
6 bnj156.3 . . . . . 6 (𝜑1[𝑔 / 𝑓]𝜑′)
76bicomi 214 . . . . 5 ([𝑔 / 𝑓]𝜑′𝜑1)
8 bnj156.4 . . . . . 6 (𝜓1[𝑔 / 𝑓]𝜓′)
98bicomi 214 . . . . 5 ([𝑔 / 𝑓]𝜓′𝜓1)
105, 7, 93anbi123i 1388 . . . 4 (([𝑔 / 𝑓]𝑓 Fn 1𝑜[𝑔 / 𝑓]𝜑′[𝑔 / 𝑓]𝜓′) ↔ (𝑔 Fn 1𝑜𝜑1𝜓1))
114, 10bitri 264 . . 3 ([𝑔 / 𝑓](𝑓 Fn 1𝑜𝜑′𝜓′) ↔ (𝑔 Fn 1𝑜𝜑1𝜓1))
123, 11bitri 264 . 2 ([𝑔 / 𝑓]𝜁0 ↔ (𝑔 Fn 1𝑜𝜑1𝜓1))
131, 12bitri 264 1 (𝜁1 ↔ (𝑔 Fn 1𝑜𝜑1𝜓1))
Colors of variables: wff setvar class
Syntax hints:  wb 196  w3a 1072  [wsbc 3541   Fn wfn 5996  1𝑜c1o 7673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-fun 6003  df-fn 6004
This theorem is referenced by:  bnj153  31178
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