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Theorem bnj156 31314
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 3689 . . 3 ([𝑔 / 𝑓]𝜁0[𝑔 / 𝑓](𝑓 Fn 1𝑜𝜑′𝜓′))
4 sbc3an 3691 . . . 4 ([𝑔 / 𝑓](𝑓 Fn 1𝑜𝜑′𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 1𝑜[𝑔 / 𝑓]𝜑′[𝑔 / 𝑓]𝜓′))
5 bnj62 31306 . . . . 5 ([𝑔 / 𝑓]𝑓 Fn 1𝑜𝑔 Fn 1𝑜)
6 bnj156.3 . . . . . 6 (𝜑1[𝑔 / 𝑓]𝜑′)
76bicomi 216 . . . . 5 ([𝑔 / 𝑓]𝜑′𝜑1)
8 bnj156.4 . . . . . 6 (𝜓1[𝑔 / 𝑓]𝜓′)
98bicomi 216 . . . . 5 ([𝑔 / 𝑓]𝜓′𝜓1)
105, 7, 93anbi123i 1195 . . . 4 (([𝑔 / 𝑓]𝑓 Fn 1𝑜[𝑔 / 𝑓]𝜑′[𝑔 / 𝑓]𝜓′) ↔ (𝑔 Fn 1𝑜𝜑1𝜓1))
114, 10bitri 267 . . 3 ([𝑔 / 𝑓](𝑓 Fn 1𝑜𝜑′𝜓′) ↔ (𝑔 Fn 1𝑜𝜑1𝜓1))
123, 11bitri 267 . 2 ([𝑔 / 𝑓]𝜁0 ↔ (𝑔 Fn 1𝑜𝜑1𝜓1))
131, 12bitri 267 1 (𝜁1 ↔ (𝑔 Fn 1𝑜𝜑1𝜓1))
Colors of variables: wff setvar class
Syntax hints:  wb 198  w3a 1108  [wsbc 3633   Fn wfn 6096  1𝑜c1o 7792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-fun 6103  df-fn 6104
This theorem is referenced by:  bnj153  31467
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