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Metamath Proof Explorer

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 ) ) ) ) )