![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-substw | Structured version Visualization version GIF version |
Description: Weak form of the LHS of bj-subst 34434 proved from the core axiom schemes. Compare ax12w 2135. (Contributed by BJ, 26-May-2024.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-substw.is | ⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-substw | ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-substw.is | . . . . 5 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓)) | |
2 | 1 | pm5.32i 579 | . . . 4 ⊢ ((𝑥 = 𝑡 ∧ 𝜑) ↔ (𝑥 = 𝑡 ∧ 𝜓)) |
3 | 2 | exbii 1850 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝑡 ∧ 𝜓)) |
4 | 19.41v 1951 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜓) ↔ (∃𝑥 𝑥 = 𝑡 ∧ 𝜓)) | |
5 | 3, 4 | bitri 278 | . 2 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ (∃𝑥 𝑥 = 𝑡 ∧ 𝜓)) |
6 | 1 | biimprcd 253 | . . 3 ⊢ (𝜓 → (𝑥 = 𝑡 → 𝜑)) |
7 | 6 | alrimiv 1929 | . 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
8 | 5, 7 | simplbiim 509 | 1 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∀wal 1537 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 |
This theorem depends on definitions: df-bi 210 df-an 401 df-ex 1783 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |