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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbval | Structured version Visualization version GIF version |
Description: Changing a bound variable (universal quantification case) in a weak axiomatization, assuming that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-cbval.denote | ⊢ ∀𝑦∃𝑥 𝑥 = 𝑦 |
bj-cbval.denote2 | ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥 |
bj-cbval.maj | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
bj-cbval.equcomiv | ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) |
Ref | Expression |
---|---|
bj-cbval | ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-cbval.maj | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | biimpd 232 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
3 | bj-cbval.denote | . . 3 ⊢ ∀𝑦∃𝑥 𝑥 = 𝑦 | |
4 | 2, 3 | bj-cbvalimi 34596 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
5 | bj-cbval.equcomiv | . . . 4 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
6 | 1 | biimprd 251 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑)) |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑)) |
8 | bj-cbval.denote2 | . . 3 ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥 | |
9 | 7, 8 | bj-cbvalimi 34596 | . 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑) |
10 | 4, 9 | impbii 212 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀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-ex 1788 |
This theorem is referenced by: (None) |
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