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Metamath Proof Explorer

Theorem cbvexd

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvexdw if possible. (Contributed by NM, 2-Jan-2002) (Revised by Mario Carneiro, 6-Oct-2016) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 cbvald.1 ⊢ Ⅎ 𝑦 𝜑
2 cbvald.2 ⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
3 cbvald.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 cbvald ⊢ ( 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 ) )