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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-equsexval | Structured version Visualization version GIF version |
Description: Special case of equsexv 2269 proved from Tarski, ax-10 2145 (modal5) and hba1 2301 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-equsexval.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓)) |
Ref | Expression |
---|---|
bj-equsexval | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-equsexval.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓)) | |
2 | 1 | pm5.32i 577 | . . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝑦 ∧ ∀𝑥𝜓)) |
3 | 2 | exbii 1848 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑥𝜓)) |
4 | ax6ev 1972 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
5 | bj-19.41al 33994 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑥𝜓) ↔ (∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜓)) | |
6 | 4, 5 | mpbiran 707 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑥𝜓) ↔ ∀𝑥𝜓) |
7 | 3, 6 | bitri 277 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 |
This theorem is referenced by: (None) |
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