Theorem rmobiia 2543

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

Hypothesis

Ref	Expression
rmobiia.1	⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rmobiia	⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓)

Proof of Theorem rmobiia

Step	Hyp	Ref	Expression
1		rmobiia.1	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))
2	1	pm5.32i 441	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))
3	2	mobii 1978	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
4		df-rmo 2356	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5		df-rmo 2356	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
6	3, 4, 5	3bitr4i 210	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∈ wcel 1433  ∃*wmo 1942  ∃*wrmo 2351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-eu 1944  df-mo 1945  df-rmo 2356

This theorem is referenced by:  rmobii  2544