Metamath Proof Explorer


Theorem bnj206

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

Ref Expression
Hypotheses bnj206.1 ( 𝜑′[ 𝑀 / 𝑛 ] 𝜑 )
bnj206.2 ( 𝜓′[ 𝑀 / 𝑛 ] 𝜓 )
bnj206.3 ( 𝜒′[ 𝑀 / 𝑛 ] 𝜒 )
bnj206.4 𝑀 ∈ V
Assertion bnj206 ( [ 𝑀 / 𝑛 ] ( 𝜑𝜓𝜒 ) ↔ ( 𝜑′𝜓′𝜒′ ) )

Proof

Step Hyp Ref Expression
1 bnj206.1 ( 𝜑′[ 𝑀 / 𝑛 ] 𝜑 )
2 bnj206.2 ( 𝜓′[ 𝑀 / 𝑛 ] 𝜓 )
3 bnj206.3 ( 𝜒′[ 𝑀 / 𝑛 ] 𝜒 )
4 bnj206.4 𝑀 ∈ V
5 sbc3an ( [ 𝑀 / 𝑛 ] ( 𝜑𝜓𝜒 ) ↔ ( [ 𝑀 / 𝑛 ] 𝜑[ 𝑀 / 𝑛 ] 𝜓[ 𝑀 / 𝑛 ] 𝜒 ) )
6 1 bicomi ( [ 𝑀 / 𝑛 ] 𝜑𝜑′ )
7 2 bicomi ( [ 𝑀 / 𝑛 ] 𝜓𝜓′ )
8 3 bicomi ( [ 𝑀 / 𝑛 ] 𝜒𝜒′ )
9 6 7 8 3anbi123i ( ( [ 𝑀 / 𝑛 ] 𝜑[ 𝑀 / 𝑛 ] 𝜓[ 𝑀 / 𝑛 ] 𝜒 ) ↔ ( 𝜑′𝜓′𝜒′ ) )
10 5 9 bitri ( [ 𝑀 / 𝑛 ] ( 𝜑𝜓𝜒 ) ↔ ( 𝜑′𝜓′𝜒′ ) )