Theorem birexd 1218

Description: Formula-building rule for restricted existential quantifier (deduction rule).

Hypotheses

Ref	Expression
birald.1	⊢ (φ → ∀xφ)
birald.2	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
birexd	⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ A χ))

Proof of Theorem birexd

Step	Hyp	Ref	Expression
1		birald.1	. 2 ⊢ (φ → ∀xφ)
2		birald.2	. . 3 ⊢ (φ → (ψ ↔ χ))
3	2	adantr 306	. 2 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
4	1, 3	birexda 1214	1 ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ A χ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  birexdv 1220  birex 1224  tz9.13g 3508  scott0 3542

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-rex 1206

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