Theorem axreplem

Description: Lemma for axrep2 and axrep3 . (Contributed by BJ, 6-Aug-2022)

	Ref	Expression
Assertion	axreplem	⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑢 ( 𝜑 → ∀ 𝑣 ( 𝜓 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑥 ∧ 𝜒 ) ) ) ↔ ∃ 𝑢 ( 𝜑 → ∀ 𝑣 ( 𝜓 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑦 ∧ 𝜒 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		elequ2	⊢ ( 𝑥 = 𝑦 → ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) )
2	1	anbi1d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑧 ∈ 𝑥 ∧ 𝜒 ) ↔ ( 𝑧 ∈ 𝑦 ∧ 𝜒 ) ) )
3	2	exbidv	⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑤 ( 𝑧 ∈ 𝑥 ∧ 𝜒 ) ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑦 ∧ 𝜒 ) ) )
4	3	bibi2d	⊢ ( 𝑥 = 𝑦 → ( ( 𝜓 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑥 ∧ 𝜒 ) ) ↔ ( 𝜓 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑦 ∧ 𝜒 ) ) ) )
5	4	albidv	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑣 ( 𝜓 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑥 ∧ 𝜒 ) ) ↔ ∀ 𝑣 ( 𝜓 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑦 ∧ 𝜒 ) ) ) )
6	5	imbi2d	⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → ∀ 𝑣 ( 𝜓 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑥 ∧ 𝜒 ) ) ) ↔ ( 𝜑 → ∀ 𝑣 ( 𝜓 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑦 ∧ 𝜒 ) ) ) ) )
7	6	exbidv	⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑢 ( 𝜑 → ∀ 𝑣 ( 𝜓 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑥 ∧ 𝜒 ) ) ) ↔ ∃ 𝑢 ( 𝜑 → ∀ 𝑣 ( 𝜓 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑦 ∧ 𝜒 ) ) ) ) )