Theorem vtocl2dOLD

Description: Obsolete version of vtocl2d as of 19-Oct-2023. (Contributed by Thierry Arnoux, 25-Aug-2020) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	vtocl2dOLD.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	vtocl2dOLD.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
	vtocl2dOLD.1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜓 ↔ 𝜒 ) )
	vtocl2dOLD.3	⊢ ( 𝜑 → 𝜓 )
Assertion	vtocl2dOLD	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		vtocl2dOLD.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		vtocl2dOLD.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3		vtocl2dOLD.1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜓 ↔ 𝜒 ) )
4		vtocl2dOLD.3	⊢ ( 𝜑 → 𝜓 )
5		nfcv	⊢ Ⅎ 𝑦 𝐵
6		nfcv	⊢ Ⅎ 𝑥 𝐵
7		nfcv	⊢ Ⅎ 𝑥 𝐴
8		nfv	⊢ Ⅎ 𝑦 𝜑
9		nfsbc1v	⊢ Ⅎ 𝑦 [ 𝐵 / 𝑦 ] 𝜓
10	8 9	nfim	⊢ Ⅎ 𝑦 ( 𝜑 → [ 𝐵 / 𝑦 ] 𝜓 )
11		nfv	⊢ Ⅎ 𝑥 ( 𝜑 → 𝜒 )
12		sbceq1a	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ [ 𝐵 / 𝑦 ] 𝜓 ) )
13	12	imbi2d	⊢ ( 𝑦 = 𝐵 → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝐵 / 𝑦 ] 𝜓 ) ) )
14		sbceq1a	⊢ ( 𝑥 = 𝐴 → ( [ 𝐵 / 𝑦 ] 𝜓 ↔ [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜓 ) )
15		nfv	⊢ Ⅎ 𝑥 𝜒
16		nfv	⊢ Ⅎ 𝑦 𝜒
17		nfv	⊢ Ⅎ 𝑥 𝐵 ∈ 𝑊
18	15 16 17 3	sbc2iegf	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜓 ↔ 𝜒 ) )
19	1 2 18	syl2anc	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜓 ↔ 𝜒 ) )
20	14 19	sylan9bb	⊢ ( ( 𝑥 = 𝐴 ∧ 𝜑 ) → ( [ 𝐵 / 𝑦 ] 𝜓 ↔ 𝜒 ) )
21	20	pm5.74da	⊢ ( 𝑥 = 𝐴 → ( ( 𝜑 → [ 𝐵 / 𝑦 ] 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) )
22	5 6 7 10 11 13 21 4	vtocl2gf	⊢ ( ( 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ) → ( 𝜑 → 𝜒 ) )
23	2 1 22	syl2anc	⊢ ( 𝜑 → ( 𝜑 → 𝜒 ) )
24	23	pm2.43i	⊢ ( 𝜑 → 𝜒 )