Theorem birexda 1214

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

Hypotheses

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

Assertion

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

Proof of Theorem birexda

Step	Hyp	Ref	Expression
1		biralda.1	. . 3 ⊢ (φ → ∀xφ)
2		biralda.2	. . . . 5 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
3	2	exp 291	. . . 4 ⊢ (φ → (x ∈ A → (ψ ↔ χ)))
4	3	pm5.32d 491	. . 3 ⊢ (φ → ((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ)))
5	1, 4	biexd 783	. 2 ⊢ (φ → (∃x(x ∈ A ∧ ψ) ↔ ∃x(x ∈ A ∧ χ)))
6		df-rex 1206	. 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ))
7		df-rex 1206	. 2 ⊢ (∃x ∈ A χ ↔ ∃x(x ∈ A ∧ χ))
8	5, 6, 7	3bitr4g 428	1 ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ A χ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  birexdva 1216  birexd 1218

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

metamath.org