Theorem gencl

Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996)

	Ref	Expression
Hypotheses	gencl.1	⊢ ( 𝜃 ↔ ∃ 𝑥 ( 𝜒 ∧ 𝐴 = 𝐵 ) )
	gencl.2	⊢ ( 𝐴 = 𝐵 → ( 𝜑 ↔ 𝜓 ) )
	gencl.3	⊢ ( 𝜒 → 𝜑 )
Assertion	gencl	⊢ ( 𝜃 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		gencl.1	⊢ ( 𝜃 ↔ ∃ 𝑥 ( 𝜒 ∧ 𝐴 = 𝐵 ) )
2		gencl.2	⊢ ( 𝐴 = 𝐵 → ( 𝜑 ↔ 𝜓 ) )
3		gencl.3	⊢ ( 𝜒 → 𝜑 )
4	3 2	syl5ib	⊢ ( 𝐴 = 𝐵 → ( 𝜒 → 𝜓 ) )
5	4	impcom	⊢ ( ( 𝜒 ∧ 𝐴 = 𝐵 ) → 𝜓 )
6	5	exlimiv	⊢ ( ∃ 𝑥 ( 𝜒 ∧ 𝐴 = 𝐵 ) → 𝜓 )
7	1 6	sylbi	⊢ ( 𝜃 → 𝜓 )