Theorem rmobida 2684

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)

Hypotheses

Ref	Expression
rmobida.1	⊢ Ⅎ𝑥𝜑
rmobida.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))

Assertion

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

Proof of Theorem rmobida

Step	Hyp	Ref	Expression
1		rmobida.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		rmobida.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
3	2	pm5.32da 452	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
4	1, 3	mobid 2080	. 2 ⊢ (𝜑 → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
5		df-rmo 2483	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
6		df-rmo 2483	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜒 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))
7	4, 5, 6	3bitr4g 223	1 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  Ⅎwnf 1474  ∃*wmo 2046   ∈ wcel 2167  ∃*wrmo 2478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-eu 2048  df-mo 2049  df-rmo 2483

This theorem is referenced by:  rmobidva  2685