Theorem rexbida 1705

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

Hypotheses

Ref	Expression
ralbida.1	⊢ (φ → ∀xφ)
ralbida.2	⊢ ((φ ⋀ x ∈ A) → (ψ ↔ χ))

Assertion

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

Proof of Theorem rexbida

Step	Hyp	Ref	Expression
1		ralbida.1	. . 3 ⊢ (φ → ∀xφ)
2		ralbida.2	. . . 4 ⊢ ((φ ⋀ x ∈ A) → (ψ ↔ χ))
3	2	pm5.32da 660	. . 3 ⊢ (φ → ((x ∈ A ⋀ ψ) ↔ (x ∈ A ⋀ χ)))
4	1, 3	exbid 1146	. 2 ⊢ (φ → (∃x(x ∈ A ⋀ ψ) ↔ ∃x(x ∈ A ⋀ χ)))
5		df-rex 1697	. 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ⋀ ψ))
6		df-rex 1697	. 2 ⊢ (∃x ∈ A χ ↔ ∃x(x ∈ A ⋀ χ))
7	4, 5, 6	3bitr4g 566	1 ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ A χ))

Syntax hints:   → wi 3   ↔ wb 153   ⋀ wa 230  ∀wal 995   ∈ wcel 999  ∃wex 1021  ∃wrex 1693

This theorem is referenced by:  rexbidva 1707  rexbid 1709  elrnopabg 3857  elrnoprabg 4182

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1004  ax-4 1014  ax-5o 1016

This theorem depends on definitions:  df-bi 154  df-an 232  df-ex 1022  df-rex 1697

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