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Mirrors > Home > MPE Home > Th. List > 19.31v | Structured version Visualization version GIF version |
Description: Version of 19.31 2234 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.) |
Ref | Expression |
---|---|
19.31v | ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.32v 1948 | . 2 ⊢ (∀𝑥(𝜓 ∨ 𝜑) ↔ (𝜓 ∨ ∀𝑥𝜑)) | |
2 | orcom 870 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑)) | |
3 | 2 | albii 1827 | . 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(𝜓 ∨ 𝜑)) |
4 | orcom 870 | . 2 ⊢ ((∀𝑥𝜑 ∨ 𝜓) ↔ (𝜓 ∨ ∀𝑥𝜑)) | |
5 | 1, 3, 4 | 3bitr4i 306 | 1 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∨ wo 847 ∀wal 1541 |
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-or 848 df-ex 1788 |
This theorem is referenced by: 19.31vv 41616 |
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