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Metamath Proof Explorer

Theorem bnj133

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj133.1 ⊢ ( 𝜑 ↔ ∃ 𝑥 𝜓 )
bnj133.2 ⊢ ( 𝜒 ↔ 𝜓 )
Assertion bnj133 ( 𝜑 ↔ ∃ 𝑥 𝜒 )

Proof

Step Hyp Ref Expression
1 bnj133.1 ⊢ ( 𝜑 ↔ ∃ 𝑥 𝜓 )
2 bnj133.2 ⊢ ( 𝜒 ↔ 𝜓 )
3 2 exbii ⊢ ( ∃ 𝑥 𝜒 ↔ ∃ 𝑥 𝜓 )
4 1 3 bitr4i ⊢ ( 𝜑 ↔ ∃ 𝑥 𝜒 )