Description: Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007) Avoid axioms. (Revised by GG, 19-May-2025)
Ref | Expression | ||
---|---|---|---|
Assertion | ssrexv | ⊢ ( 𝐴 ⊆ 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜑 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ss | ⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ) | |
2 | pm3.45 | ⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) ) | |
3 | 2 | aleximi | ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) ) |
4 | df-rex | ⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) | |
5 | df-rex | ⊢ ( ∃ 𝑥 ∈ 𝐵 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) | |
6 | 3 4 5 | 3imtr4g | ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜑 ) ) |
7 | 1 6 | sylbi | ⊢ ( 𝐴 ⊆ 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜑 ) ) |