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Metamath Proof Explorer

Theorem exbidh

Description: Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 26-May-1993)

Ref Expression
Hypotheses exbidh.1 ⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
exbidh.2 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion exbidh ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑥 𝜒 ) )

Proof

Step Hyp Ref Expression
1 exbidh.1 ⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2 exbidh.2 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3 2 alexbii ⊢ ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑥 𝜒 ) )
4 1 3 syl ⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑥 𝜒 ) )