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Metamath Proof Explorer

Theorem cbvexdw

Description: Deduction used to change bound variables, using implicit substitution. Version of cbvexd with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 2-Jan-2002) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

Ref Expression
Hypotheses cbvaldw.1 ⊢ Ⅎ 𝑦 𝜑
cbvaldw.2 ⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
cbvaldw.3 ⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
Assertion cbvexdw ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑦 𝜒 ) )

Proof

Step Hyp Ref Expression
1 cbvaldw.1 ⊢ Ⅎ 𝑦 𝜑
2 cbvaldw.2 ⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
3 cbvaldw.3 ⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
4 2 nfnd ⊢ ( 𝜑 → Ⅎ 𝑦 ¬ 𝜓 )
5 notbi ⊢ ( ( 𝜓 ↔ 𝜒 ) ↔ ( ¬ 𝜓 ↔ ¬ 𝜒 ) )
6 3 5 imbitrdi ⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( ¬ 𝜓 ↔ ¬ 𝜒 ) ) )
7 1 4 6 cbvaldw ⊢ ( 𝜑 → ( ∀ 𝑥 ¬ 𝜓 ↔ ∀ 𝑦 ¬ 𝜒 ) )
8 alnex ⊢ ( ∀ 𝑥 ¬ 𝜓 ↔ ¬ ∃ 𝑥 𝜓 )
9 alnex ⊢ ( ∀ 𝑦 ¬ 𝜒 ↔ ¬ ∃ 𝑦 𝜒 )
10 7 8 9 3bitr3g ⊢ ( 𝜑 → ( ¬ ∃ 𝑥 𝜓 ↔ ¬ ∃ 𝑦 𝜒 ) )
11 10 con4bid ⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑦 𝜒 ) )