Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (Proof shortened by Mario Carneiro, 22-Dec-2016) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypothesis | bnj90.1 | ⊢ 𝑌 ∈ V | |
Assertion | bnj90 | ⊢ ( [ 𝑌 / 𝑥 ] 𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj90.1 | ⊢ 𝑌 ∈ V | |
2 | fneq2 | ⊢ ( 𝑥 = 𝑦 → ( 𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑦 ) ) | |
3 | fneq2 | ⊢ ( 𝑦 = 𝑌 → ( 𝑧 Fn 𝑦 ↔ 𝑧 Fn 𝑌 ) ) | |
4 | 2 3 | sbcie2g | ⊢ ( 𝑌 ∈ V → ( [ 𝑌 / 𝑥 ] 𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌 ) ) |
5 | 1 4 | ax-mp | ⊢ ( [ 𝑌 / 𝑥 ] 𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌 ) |