Theorem rmoimi2 2933

Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Hypothesis

Ref	Expression
rmoimi2.1	⊢ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜓))

Assertion

Ref	Expression
rmoimi2	⊢ (∃*𝑥 ∈ 𝐵 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)

Proof of Theorem rmoimi2

Step	Hyp	Ref	Expression
1		rmoimi2.1	. . 3 ⊢ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜓))
2		moim 2083	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜓)) → (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
3	1, 2	ax-mp 5	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-rmo 2456	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
5		df-rmo 2456	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6	3, 4, 5	3imtr4i 200	1 ⊢ (∃*𝑥 ∈ 𝐵 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1346  ∃*wmo 2020   ∈ wcel 2141  ∃*wrmo 2451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-rmo 2456

This theorem is referenced by: (None)