Theorem bnj1465

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1465.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	bnj1465.2	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
	bnj1465.3	⊢ ( 𝜒 → 𝜓 )
Assertion	bnj1465	⊢ ( ( 𝜒 ∧ 𝐴 ∈ 𝑉 ) → ∃ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		bnj1465.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		bnj1465.2	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
3		bnj1465.3	⊢ ( 𝜒 → 𝜓 )
4	3	adantr	⊢ ( ( 𝜒 ∧ 𝐴 ∈ 𝑉 ) → 𝜓 )
5	2 1	bnj1464	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜓 ) )
6	5	adantl	⊢ ( ( 𝜒 ∧ 𝐴 ∈ 𝑉 ) → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜓 ) )
7	4 6	mpbird	⊢ ( ( 𝜒 ∧ 𝐴 ∈ 𝑉 ) → [ 𝐴 / 𝑥 ] 𝜑 )
8	7	spesbcd	⊢ ( ( 𝜒 ∧ 𝐴 ∈ 𝑉 ) → ∃ 𝑥 𝜑 )