Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypotheses | bnj206.1 | ⊢ ( 𝜑′ ↔ [ 𝑀 / 𝑛 ] 𝜑 ) | |
bnj206.2 | ⊢ ( 𝜓′ ↔ [ 𝑀 / 𝑛 ] 𝜓 ) | ||
bnj206.3 | ⊢ ( 𝜒′ ↔ [ 𝑀 / 𝑛 ] 𝜒 ) | ||
bnj206.4 | ⊢ 𝑀 ∈ V | ||
Assertion | bnj206 | ⊢ ( [ 𝑀 / 𝑛 ] ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( 𝜑′ ∧ 𝜓′ ∧ 𝜒′ ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj206.1 | ⊢ ( 𝜑′ ↔ [ 𝑀 / 𝑛 ] 𝜑 ) | |
2 | bnj206.2 | ⊢ ( 𝜓′ ↔ [ 𝑀 / 𝑛 ] 𝜓 ) | |
3 | bnj206.3 | ⊢ ( 𝜒′ ↔ [ 𝑀 / 𝑛 ] 𝜒 ) | |
4 | bnj206.4 | ⊢ 𝑀 ∈ V | |
5 | sbc3an | ⊢ ( [ 𝑀 / 𝑛 ] ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( [ 𝑀 / 𝑛 ] 𝜑 ∧ [ 𝑀 / 𝑛 ] 𝜓 ∧ [ 𝑀 / 𝑛 ] 𝜒 ) ) | |
6 | 1 | bicomi | ⊢ ( [ 𝑀 / 𝑛 ] 𝜑 ↔ 𝜑′ ) |
7 | 2 | bicomi | ⊢ ( [ 𝑀 / 𝑛 ] 𝜓 ↔ 𝜓′ ) |
8 | 3 | bicomi | ⊢ ( [ 𝑀 / 𝑛 ] 𝜒 ↔ 𝜒′ ) |
9 | 6 7 8 | 3anbi123i | ⊢ ( ( [ 𝑀 / 𝑛 ] 𝜑 ∧ [ 𝑀 / 𝑛 ] 𝜓 ∧ [ 𝑀 / 𝑛 ] 𝜒 ) ↔ ( 𝜑′ ∧ 𝜓′ ∧ 𝜒′ ) ) |
10 | 5 9 | bitri | ⊢ ( [ 𝑀 / 𝑛 ] ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( 𝜑′ ∧ 𝜓′ ∧ 𝜒′ ) ) |