Theorem sbcie2s

Description: A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019) (Revised by SN, 2-Mar-2025)

	Ref	Expression
Hypotheses	sbcie2s.a	⊢ 𝐴 = ( 𝐸 ‘ 𝑊 )
	sbcie2s.b	⊢ 𝐵 = ( 𝐹 ‘ 𝑊 )
	sbcie2s.1	⊢ ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	sbcie2s	⊢ ( 𝑤 = 𝑊 → ( [ ( 𝐸 ‘ 𝑤 ) / 𝑎 ] [ ( 𝐹 ‘ 𝑤 ) / 𝑏 ] 𝜑 ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		sbcie2s.a	⊢ 𝐴 = ( 𝐸 ‘ 𝑊 )
2		sbcie2s.b	⊢ 𝐵 = ( 𝐹 ‘ 𝑊 )
3		sbcie2s.1	⊢ ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
4		fvex	⊢ ( 𝐸 ‘ 𝑤 ) ∈ V
5		fvex	⊢ ( 𝐹 ‘ 𝑤 ) ∈ V
6		fveq2	⊢ ( 𝑤 = 𝑊 → ( 𝐸 ‘ 𝑤 ) = ( 𝐸 ‘ 𝑊 ) )
7	6 1	eqtr4di	⊢ ( 𝑤 = 𝑊 → ( 𝐸 ‘ 𝑤 ) = 𝐴 )
8	7	eqeq2d	⊢ ( 𝑤 = 𝑊 → ( 𝑎 = ( 𝐸 ‘ 𝑤 ) ↔ 𝑎 = 𝐴 ) )
9	8	biimpd	⊢ ( 𝑤 = 𝑊 → ( 𝑎 = ( 𝐸 ‘ 𝑤 ) → 𝑎 = 𝐴 ) )
10		fveq2	⊢ ( 𝑤 = 𝑊 → ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑊 ) )
11	10 2	eqtr4di	⊢ ( 𝑤 = 𝑊 → ( 𝐹 ‘ 𝑤 ) = 𝐵 )
12	11	eqeq2d	⊢ ( 𝑤 = 𝑊 → ( 𝑏 = ( 𝐹 ‘ 𝑤 ) ↔ 𝑏 = 𝐵 ) )
13	12	biimpd	⊢ ( 𝑤 = 𝑊 → ( 𝑏 = ( 𝐹 ‘ 𝑤 ) → 𝑏 = 𝐵 ) )
14	3	a1i	⊢ ( 𝑤 = 𝑊 → ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) )
15	9 13 14	syl2and	⊢ ( 𝑤 = 𝑊 → ( ( 𝑎 = ( 𝐸 ‘ 𝑤 ) ∧ 𝑏 = ( 𝐹 ‘ 𝑤 ) ) → ( 𝜑 ↔ 𝜓 ) ) )
16	4 5 15	sbc2iedv	⊢ ( 𝑤 = 𝑊 → ( [ ( 𝐸 ‘ 𝑤 ) / 𝑎 ] [ ( 𝐹 ‘ 𝑤 ) / 𝑏 ] 𝜑 ↔ 𝜓 ) )