Metamath Proof Explorer


Theorem bnj984

Description: Technical lemma for bnj69 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj984.3 ( 𝜒 ↔ ( 𝑛𝐷𝑓 Fn 𝑛𝜑𝜓 ) )
bnj984.11 𝐵 = { 𝑓 ∣ ∃ 𝑛𝐷 ( 𝑓 Fn 𝑛𝜑𝜓 ) }
Assertion bnj984 ( 𝐺𝐴 → ( 𝐺𝐵[ 𝐺 / 𝑓 ]𝑛 𝜒 ) )

Proof

Step Hyp Ref Expression
1 bnj984.3 ( 𝜒 ↔ ( 𝑛𝐷𝑓 Fn 𝑛𝜑𝜓 ) )
2 bnj984.11 𝐵 = { 𝑓 ∣ ∃ 𝑛𝐷 ( 𝑓 Fn 𝑛𝜑𝜓 ) }
3 2 eleq2i ( 𝐺𝐵𝐺 ∈ { 𝑓 ∣ ∃ 𝑛𝐷 ( 𝑓 Fn 𝑛𝜑𝜓 ) } )
4 sbc8g ( 𝐺𝐴 → ( [ 𝐺 / 𝑓 ]𝑛𝐷 ( 𝑓 Fn 𝑛𝜑𝜓 ) ↔ 𝐺 ∈ { 𝑓 ∣ ∃ 𝑛𝐷 ( 𝑓 Fn 𝑛𝜑𝜓 ) } ) )
5 3 4 bitr4id ( 𝐺𝐴 → ( 𝐺𝐵[ 𝐺 / 𝑓 ]𝑛𝐷 ( 𝑓 Fn 𝑛𝜑𝜓 ) ) )
6 df-rex ( ∃ 𝑛𝐷 ( 𝑓 Fn 𝑛𝜑𝜓 ) ↔ ∃ 𝑛 ( 𝑛𝐷 ∧ ( 𝑓 Fn 𝑛𝜑𝜓 ) ) )
7 bnj252 ( ( 𝑛𝐷𝑓 Fn 𝑛𝜑𝜓 ) ↔ ( 𝑛𝐷 ∧ ( 𝑓 Fn 𝑛𝜑𝜓 ) ) )
8 1 7 bitri ( 𝜒 ↔ ( 𝑛𝐷 ∧ ( 𝑓 Fn 𝑛𝜑𝜓 ) ) )
9 6 8 bnj133 ( ∃ 𝑛𝐷 ( 𝑓 Fn 𝑛𝜑𝜓 ) ↔ ∃ 𝑛 𝜒 )
10 9 sbcbii ( [ 𝐺 / 𝑓 ]𝑛𝐷 ( 𝑓 Fn 𝑛𝜑𝜓 ) ↔ [ 𝐺 / 𝑓 ]𝑛 𝜒 )
11 5 10 bitrdi ( 𝐺𝐴 → ( 𝐺𝐵[ 𝐺 / 𝑓 ]𝑛 𝜒 ) )