Metamath Proof Explorer


Theorem cdleme31sdnN

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 31-Mar-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme31sdn.c 𝐶 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
cdleme31sdn.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
cdleme31sdn.n 𝑁 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝐶 )
Assertion cdleme31sdnN 𝑁 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝑠 / 𝑡 𝐷 )

Proof

Step Hyp Ref Expression
1 cdleme31sdn.c 𝐶 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
2 cdleme31sdn.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
3 cdleme31sdn.n 𝑁 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝐶 )
4 biid ( 𝑠 ( 𝑃 𝑄 ) ↔ 𝑠 ( 𝑃 𝑄 ) )
5 2 1 cdleme31sc ( 𝑠 ∈ V → 𝑠 / 𝑡 𝐷 = 𝐶 )
6 5 elv 𝑠 / 𝑡 𝐷 = 𝐶
7 4 6 ifbieq2i if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝑠 / 𝑡 𝐷 ) = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝐶 )
8 3 7 eqtr4i 𝑁 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝑠 / 𝑡 𝐷 )