Metamath Proof Explorer


Theorem bnj156

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses 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 bnj156 ( 𝜁1 ↔ ( 𝑔 Fn 1o𝜑1𝜓1 ) )

Proof

Step Hyp Ref Expression
1 bnj156.1 ( 𝜁0 ↔ ( 𝑓 Fn 1o𝜑′𝜓′ ) )
2 bnj156.2 ( 𝜁1[ 𝑔 / 𝑓 ] 𝜁0 )
3 bnj156.3 ( 𝜑1[ 𝑔 / 𝑓 ] 𝜑′ )
4 bnj156.4 ( 𝜓1[ 𝑔 / 𝑓 ] 𝜓′ )
5 1 sbcbii ( [ 𝑔 / 𝑓 ] 𝜁0[ 𝑔 / 𝑓 ] ( 𝑓 Fn 1o𝜑′𝜓′ ) )
6 sbc3an ( [ 𝑔 / 𝑓 ] ( 𝑓 Fn 1o𝜑′𝜓′ ) ↔ ( [ 𝑔 / 𝑓 ] 𝑓 Fn 1o[ 𝑔 / 𝑓 ] 𝜑′[ 𝑔 / 𝑓 ] 𝜓′ ) )
7 bnj62 ( [ 𝑔 / 𝑓 ] 𝑓 Fn 1o𝑔 Fn 1o )
8 3 bicomi ( [ 𝑔 / 𝑓 ] 𝜑′𝜑1 )
9 4 bicomi ( [ 𝑔 / 𝑓 ] 𝜓′𝜓1 )
10 7 8 9 3anbi123i ( ( [ 𝑔 / 𝑓 ] 𝑓 Fn 1o[ 𝑔 / 𝑓 ] 𝜑′[ 𝑔 / 𝑓 ] 𝜓′ ) ↔ ( 𝑔 Fn 1o𝜑1𝜓1 ) )
11 6 10 bitri ( [ 𝑔 / 𝑓 ] ( 𝑓 Fn 1o𝜑′𝜓′ ) ↔ ( 𝑔 Fn 1o𝜑1𝜓1 ) )
12 5 11 bitri ( [ 𝑔 / 𝑓 ] 𝜁0 ↔ ( 𝑔 Fn 1o𝜑1𝜓1 ) )
13 2 12 bitri ( 𝜁1 ↔ ( 𝑔 Fn 1o𝜑1𝜓1 ) )