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Theorem bnj156 32239
 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 3755 . . 3 ([𝑔 / 𝑓]𝜁0[𝑔 / 𝑓](𝑓 Fn 1o𝜑′𝜓′))
4 sbc3an 3764 . . . 4 ([𝑔 / 𝑓](𝑓 Fn 1o𝜑′𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 1o[𝑔 / 𝑓]𝜑′[𝑔 / 𝑓]𝜓′))
5 bnj62 32231 . . . . 5 ([𝑔 / 𝑓]𝑓 Fn 1o𝑔 Fn 1o)
6 bnj156.3 . . . . . 6 (𝜑1[𝑔 / 𝑓]𝜑′)
76bicomi 227 . . . . 5 ([𝑔 / 𝑓]𝜑′𝜑1)
8 bnj156.4 . . . . . 6 (𝜓1[𝑔 / 𝑓]𝜓′)
98bicomi 227 . . . . 5 ([𝑔 / 𝑓]𝜓′𝜓1)
105, 7, 93anbi123i 1152 . . . 4 (([𝑔 / 𝑓]𝑓 Fn 1o[𝑔 / 𝑓]𝜑′[𝑔 / 𝑓]𝜓′) ↔ (𝑔 Fn 1o𝜑1𝜓1))
114, 10bitri 278 . . 3 ([𝑔 / 𝑓](𝑓 Fn 1o𝜑′𝜓′) ↔ (𝑔 Fn 1o𝜑1𝜓1))
123, 11bitri 278 . 2 ([𝑔 / 𝑓]𝜁0 ↔ (𝑔 Fn 1o𝜑1𝜓1))
131, 12bitri 278 1 (𝜁1 ↔ (𝑔 Fn 1o𝜑1𝜓1))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ w3a 1084  [wsbc 3698   Fn wfn 6335  1oc1o 8111 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-10 2142  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-sbc 3699  df-un 3865  df-in 3867  df-ss 3877  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-fun 6342  df-fn 6343 This theorem is referenced by:  bnj153  32393
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