Theorem rdgsucmpt2

Description: This version of rdgsucmpt avoids the not-free hypothesis of rdgsucmptf by using two substitutions instead of one. (Contributed by Mario Carneiro, 11-Sep-2015)

	Ref	Expression
Hypotheses	rdgsucmpt2.1	⊢ 𝐹 = rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 )
	rdgsucmpt2.2	⊢ ( 𝑦 = 𝑥 → 𝐸 = 𝐶 )
	rdgsucmpt2.3	⊢ ( 𝑦 = ( 𝐹 ‘ 𝐵 ) → 𝐸 = 𝐷 )
Assertion	rdgsucmpt2	⊢ ( ( 𝐵 ∈ On ∧ 𝐷 ∈ 𝑉 ) → ( 𝐹 ‘ suc 𝐵 ) = 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		rdgsucmpt2.1	⊢ 𝐹 = rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 )
2		rdgsucmpt2.2	⊢ ( 𝑦 = 𝑥 → 𝐸 = 𝐶 )
3		rdgsucmpt2.3	⊢ ( 𝑦 = ( 𝐹 ‘ 𝐵 ) → 𝐸 = 𝐷 )
4		nfcv	⊢ Ⅎ 𝑦 𝐴
5		nfcv	⊢ Ⅎ 𝑦 𝐵
6		nfcv	⊢ Ⅎ 𝑦 𝐷
7	2	cbvmptv	⊢ ( 𝑦 ∈ V ↦ 𝐸 ) = ( 𝑥 ∈ V ↦ 𝐶 )
8		rdgeq1	⊢ ( ( 𝑦 ∈ V ↦ 𝐸 ) = ( 𝑥 ∈ V ↦ 𝐶 ) → rec ( ( 𝑦 ∈ V ↦ 𝐸 ) , 𝐴 ) = rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 ) )
9	7 8	ax-mp	⊢ rec ( ( 𝑦 ∈ V ↦ 𝐸 ) , 𝐴 ) = rec ( ( 𝑥 ∈ V ↦ 𝐶 ) , 𝐴 )
10	1 9	eqtr4i	⊢ 𝐹 = rec ( ( 𝑦 ∈ V ↦ 𝐸 ) , 𝐴 )
11	4 5 6 10 3	rdgsucmptf	⊢ ( ( 𝐵 ∈ On ∧ 𝐷 ∈ 𝑉 ) → ( 𝐹 ‘ suc 𝐵 ) = 𝐷 )