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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-substw | Structured version Visualization version GIF version |
Description: Weak form of the LHS of bj-substax12 34640 proved from the core axiom schemes. Compare ax12w 2133. (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 578 | . . . 4 ⊢ ((𝑥 = 𝑡 ∧ 𝜑) ↔ (𝑥 = 𝑡 ∧ 𝜓)) |
3 | 2 | exbii 1855 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝑡 ∧ 𝜓)) |
4 | 19.41v 1958 | . . 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 1935 | . 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
8 | 5, 7 | simplbiim 508 | 1 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 ∃wex 1787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 |
This theorem is referenced by: (None) |
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