Description: Alternate elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020)
Ref | Expression | ||
---|---|---|---|
Hypothesis | ceqsralv2.1 | ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) | |
Assertion | ceqsralv2 | ⊢ ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ( 𝐴 ∈ 𝐵 → 𝜓 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ceqsralv2.1 | ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) | |
2 | 1 | notbid | ⊢ ( 𝑥 = 𝐴 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) ) |
3 | 2 | ceqsrexv2 | ⊢ ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 ∧ ¬ 𝜑 ) ↔ ( 𝐴 ∈ 𝐵 ∧ ¬ 𝜓 ) ) |
4 | rexanali | ⊢ ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 ∧ ¬ 𝜑 ) ↔ ¬ ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ) | |
5 | annim | ⊢ ( ( 𝐴 ∈ 𝐵 ∧ ¬ 𝜓 ) ↔ ¬ ( 𝐴 ∈ 𝐵 → 𝜓 ) ) | |
6 | 3 4 5 | 3bitr3i | ⊢ ( ¬ ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ¬ ( 𝐴 ∈ 𝐵 → 𝜓 ) ) |
7 | 6 | con4bii | ⊢ ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ( 𝐴 ∈ 𝐵 → 𝜓 ) ) |